Daily Papers

The tool-use Large Language Models (LLMs) that integrate with external Python interpreters have significantly enhanced mathematical reasoning capabilities for open-source LLMs, while tool-free methods chose another track: augmenting math reasoning data. However, a great method to integrate the above two research paths and combine their advantages remains to be explored. In this work, we firstly include new math questions via multi-perspective data augmenting methods and then synthesize code-nested solutions to them. The open LLMs (i.e., Llama-2) are finetuned on the augmented dataset to get the resulting models, MuMath-Code (mu-Math-Code). During the inference phase, our MuMath-Code generates code and interacts with the external python interpreter to get the execution results. Therefore, MuMath-Code leverages the advantages of both the external tool and data augmentation. To fully leverage the advantages of our augmented data, we propose a two-stage training strategy: In Stage-1, we finetune Llama-2 on pure CoT data to get an intermediate model, which then is trained on the code-nested data in Stage-2 to get the resulting MuMath-Code. Our MuMath-Code-7B achieves 83.8 on GSM8K and 52.4 on MATH, while MuMath-Code-70B model achieves new state-of-the-art performance among open methods -- achieving 90.7% on GSM8K and 55.1% on MATH. Extensive experiments validate the combination of tool use and data augmentation, as well as our two-stage training strategy. We release the proposed dataset along with the associated code for public use.

We present MixtureVitae, an open-access pretraining corpus built to minimize legal risk while providing strong model performance. MixtureVitae follows a risk-mitigated sourcing strategy that combines public-domain and permissively licensed text (e.g., CC-BY/Apache) with carefully justified low-risk additions (e.g., government works and EU TDM-eligible sources), alongside targeted instruction, reasoning and synthetic data with documented provenance. We detail a transparent, multi-stage pipeline for license-aware filtering, safety and quality screening, and domain-aware mixing, and we release the dataset and curation recipes to support reproducible research. In controlled experiments using the open-sci-ref training protocol (fixed architectures at 130M/400M/1.3B/1.7B parameters; training budgets of 50B and 300B tokens), models trained on MixtureVitae consistently outperform other permissive datasets across a suite of standard benchmarks, and at the 1.7B/300B setting they surpass FineWeb-Edu and approach DCLM in the later stages of training. Performance is particularly strong on math/code and competitive on QA tasks. These results demonstrate that permissive-first, risk-mitigated data provides a practical and legally mitigated foundation for training capable LLMs, reducing reliance on indiscriminate web scraping without sacrificing competitiveness. Code: https://github.com/ontocord/mixturevitae

Current parameter-efficient fine-tuning (PEFT) methods build adapters without considering the context of downstream task to learn, or the context of important knowledge to maintain. As a result, there is often a performance gap compared to full-parameter finetuning, and meanwhile the finetuned model suffers from catastrophic forgetting of the pre-trained world knowledge. In this paper, we propose CorDA, a Context-oriented Decomposition Adaptation method that builds learnable adapters from weight decomposition oriented by the context of downstream task or world knowledge. Concretely, we collect a few data samples, and perform singular value decomposition for each linear layer of a pre-trained LLM multiplied by the covariance matrix of the input activation using these samples. By doing so, the context of the representative samples is captured through deciding the factorizing orientation. Our method enables two options, the knowledge-preserved adaptation and the instruction-previewed adaptation. For the former, we use question-answering samples to obtain the covariance matrices, and use the decomposed components with the smallest r singular values to initialize a learnable adapter, with the others frozen such that the world knowledge is better preserved. For the latter, we use the instruction data from the finetuning task, such as math or coding, to orientate the decomposition and train the largest r components that capture the main characteristics of the task to learn. We conduct extensive experiments on Math, Code, and Instruction Following tasks. Our knowledge-preserved adaptation not only achieves better performance than LoRA on finetuning tasks, but also mitigates the forgetting of world knowledge. Our instruction-previewed adaptation is able to further enhance the finetuning performance, surpassing full-parameter finetuning and the state-of-the-art PEFT methods.

Reasoning models have made rapid progress on many benchmarks involving math, code, and science. Yet, there are still many open questions about the best training recipes for reasoning since state-of-the-art models often rely on proprietary datasets with little to no public information available. To address this, the goal of the OpenThoughts project is to create open-source datasets for training reasoning models. After initial explorations, our OpenThoughts2-1M dataset led to OpenThinker2-32B, the first model trained on public reasoning data to match DeepSeek-R1-Distill-32B on standard reasoning benchmarks such as AIME and LiveCodeBench. We then improve our dataset further by systematically investigating each step of our data generation pipeline with 1,000+ controlled experiments, which led to OpenThoughts3. Scaling the pipeline to 1.2M examples and using QwQ-32B as teacher yields our OpenThinker3-7B model, which achieves state-of-the-art results: 53% on AIME 2025, 51% on LiveCodeBench 06/24-01/25, and 54% on GPQA Diamond. All of our datasets and models are available on https://openthoughts.ai.

In recent years, the research focus of large language models (LLMs) and agents has shifted increasingly from demonstrating novel capabilities to complex reasoning and tackling challenging tasks. However, existing evaluations focus mainly on math/code contests or general tasks, while existing multi-domain academic benchmarks lack sufficient reasoning depth, leaving the field without a rigorous benchmark for high-level reasoning. To fill this gap, we introduce the Acadreason benchmark, designed to evaluate the ability of LLMs and agents to acquire and reason over academic knowledge. It consists of 50 expert-annotated academic problems across five high-reasoning domains, including computer science, economics, law, mathematics, and philosophy. All questions are sourced from top-tier publications in recent years and undergo rigorous annotation and quality control to ensure they are both challenging and answerable. We conduct systematic evaluations of over 10 mainstream LLMs and agents. The results show that most LLMs scored below 20 points, with even the cutting-edge GPT-5 achieving only 16 points. While agents achieved higher scores, none exceeded 40 points. This demonstrates the current capability gap between LLMs and agents in super-intelligent academic research tasks and highlights the challenges of Acadreason.

Reinforcement learning with verifiable rewards (RLVR) is a practical and scalable approach to enhancing large language models in areas such as math, code, and other structured tasks. Two questions motivate this paper: how much of the reported gains survive under strictly parity-controlled evaluation, and whether RLVR is cost-free or exacts a measurable tax. We argue that progress is real, but gains are often overstated due to three forces - an RLVR tax, evaluation pitfalls, and data contamination. Using a partial-prompt contamination audit and matched-budget reproductions across base and RL models, we show that several headline gaps shrink or vanish under clean, parity-controlled evaluation. We then propose a tax-aware training and evaluation protocol that co-optimizes accuracy, grounding, and calibrated abstention and standardizes budgeting and provenance checks. Applied to recent RLVR setups, this protocol yields more reliable estimates of reasoning gains and, in several cases, revises prior conclusions. Our position is constructive: RLVR is valuable and industry-ready; we advocate keeping its practical benefits while prioritizing reliability, safety, and measurement.

Recent advances in large language models have led to specialized models excelling in specific domains, creating a need for efficient model merging techniques. While traditional merging approaches combine parameters into a single static model, they often compromise task-specific performance. However, task-specific routing methods maintain accuracy but introduce substantial storage overhead. We present 1bit-Merging, a novel framework that integrates task-specific routing with 1-bit quantized task vectors to balance performance and storage efficiency. Our approach leverages the observation that different task-specific models store knowledge in distinct layers-chat models primarily in attention layers and math/code models in MLP layers-enabling targeted compression strategies. Through extensive experiments with LLaMA2 and Mistral model families across chat, mathematical reasoning, and code generation tasks, we demonstrate that 1bit-Merging achieves comparable or superior performance to existing methods while significantly reducing storage requirements. Our framework offers a practical solution for combining specialized models while maintaining their individual strengths and addressing the storage challenges of current approaches.

Despite recent progress in large-scale reinforcement learning (RL) for reasoning, the training recipe for building high-performing reasoning models remains elusive. Key implementation details of frontier models, such as DeepSeek-R1, including data curation strategies and RL training recipe, are often omitted. Moreover, recent research indicates distillation remains more effective than RL for smaller models. In this work, we demonstrate that large-scale RL can significantly enhance the reasoning capabilities of strong, small- and mid-sized models, achieving results that surpass those of state-of-the-art distillation-based models. We systematically study the RL training process through extensive ablations and propose a simple yet effective approach: first training on math-only prompts, then on code-only prompts. Notably, we find that math-only RL not only significantly enhances the performance of strong distilled models on math benchmarks (e.g., +14.6% / +17.2% on AIME 2025 for the 7B / 14B models), but also code reasoning tasks (e.g., +6.8% / +5.8% on LiveCodeBench for the 7B / 14B models). In addition, extended code-only RL iterations further improve performance on code benchmarks with minimal or no degradation in math results. We develop a robust data curation pipeline to collect challenging prompts with high-quality, verifiable answers and test cases to enable verification-based RL across both domains. Finally, we identify key experimental insights, including curriculum learning with progressively increasing response lengths and the stabilizing effect of on-policy parameter updates. We find that RL not only elicits the foundational reasoning capabilities acquired during pretraining and supervised fine-tuning (e.g., distillation), but also pushes the limits of the model's reasoning ability, enabling it to solve problems that were previously unsolvable.

In this work, we investigate the synergy between supervised fine-tuning (SFT) and reinforcement learning (RL) in developing strong reasoning models. We begin by curating the SFT training data through two scaling strategies: increasing the number of collected prompts and the number of generated responses per prompt. Both approaches yield notable improvements in reasoning performance, with scaling the number of prompts resulting in more substantial gains. We then explore the following questions regarding the synergy between SFT and RL: (i) Does a stronger SFT model consistently lead to better final performance after large-scale RL training? (ii) How can we determine an appropriate sampling temperature during RL training to effectively balance exploration and exploitation for a given SFT initialization? Our findings suggest that (i) holds true, provided effective RL training is conducted, particularly when the sampling temperature is carefully chosen to maintain the temperature-adjusted entropy around 0.3, a setting that strikes a good balance between exploration and exploitation. Notably, the performance gap between initial SFT models narrows significantly throughout the RL process. Leveraging a strong SFT foundation and insights into the synergistic interplay between SFT and RL, our AceReason-Nemotron-1.1 7B model significantly outperforms AceReason-Nemotron-1.0 and achieves new state-of-the-art performance among Qwen2.5-7B-based reasoning models on challenging math and code benchmarks, thereby demonstrating the effectiveness of our post-training recipe. We release the model and data at: https://huggingface.co/nvidia/AceReason-Nemotron-1.1-7B

Large language models (LLMs) have demonstrated remarkable reasoning capabilities in math and coding, often bolstered by post-training on the chain-of-thoughts (CoTs) generated by stronger models. However, existing strategies for curating such training data predominantly rely on heuristics, limiting generalizability and failing to capture subtleties underlying in data. To address these limitations, we leverage influence functions to systematically attribute LLMs' reasoning ability on math and coding to individual training examples, sequences, and tokens, enabling deeper insights into effective data characteristics. Our Influence-based Reasoning Attribution (Infra) uncovers nontrivial cross-domain effects across math and coding tasks: high-difficulty math examples improve both math and code reasoning, while low-difficulty code tasks most effectively benefit code reasoning. Based on these findings, we introduce a simple yet effective dataset reweighting strategy by flipping task difficulty, which doubles AIME24 accuracy from 10\% to 20\% and boosts LiveCodeBench accuracy from 33.8\% to 35.3\% for Qwen2.5-7B-Instruct. Moreover, our fine-grained attribution reveals that the sequence-level exploratory behaviors enhance reasoning performance in both math and code, and the token-level influence patterns are distinct for math and code reasoning: the former prefers natural language logic connectors and the latter emphasizes structural syntax.

The performance of large language models (LLMs) in program synthesis and mathematical reasoning is fundamentally limited by the quality of their pre-training corpora. We introduce two openly licensed datasets, released under the Llama 3.3 Community License, that significantly enhance LLM performance by systematically rewriting public data. SwallowCode (approximately 16.1 billion tokens) refines Python snippets from The-Stack-v2 through a novel four-stage pipeline: syntax validation, pylint-based style filtering, and a two-stage LLM rewriting process that enforces style conformity and transforms snippets into self-contained, algorithmically efficient examples. Unlike prior methods that rely on exclusionary filtering or limited transformations, our transform-and-retain approach upgrades low-quality code, maximizing data utility. SwallowMath (approximately 2.3 billion tokens) enhances Finemath-4+ by removing boilerplate, restoring context, and reformatting solutions into concise, step-by-step explanations. Within a fixed 50 billion token training budget, continual pre-training of Llama-3.1-8B with SwallowCode boosts pass@1 by +17.0 on HumanEval and +17.7 on HumanEval+ compared to Stack-Edu, surpassing the baseline model's code generation capabilities. Similarly, substituting SwallowMath yields +12.4 accuracy on GSM8K and +7.6 on MATH. Ablation studies confirm that each pipeline stage contributes incrementally, with rewriting delivering the largest gains. All datasets, prompts, and checkpoints are publicly available, enabling reproducible research and advancing LLM pre-training for specialized domains.

Mathematical reasoning is a cornerstone of human intelligence and a key benchmark for advanced capabilities in large language models (LLMs). However, the research community still lacks an open, large-scale, high-quality corpus tailored to the demands of math-centric LLM pre-training. We present MegaMath, an open dataset curated from diverse, math-focused sources through following practices: (1) Revisiting web data: We re-extracted mathematical documents from Common Crawl with math-oriented HTML optimizations, fasttext-based filtering and deduplication, all for acquiring higher-quality data on the Internet. (2) Recalling Math-related code data: We identified high quality math-related code from large code training corpus, Stack-V2, further enhancing data diversity. (3) Exploring Synthetic data: We synthesized QA-style text, math-related code, and interleaved text-code blocks from web data or code data. By integrating these strategies and validating their effectiveness through extensive ablations, MegaMath delivers 371B tokens with the largest quantity and top quality among existing open math pre-training datasets.

Reasoning is a fundamental capability of Large Language Models. While prior research predominantly focuses on enhancing narrow skills like math or code generation, improving performance on many other reasoning tasks remains challenging due to sparse and fragmented training data. To address this issue, we propose CodeI/O, a novel approach that systematically condenses diverse reasoning patterns inherently embedded in contextually-grounded codes, through transforming the original code into a code input-output prediction format. By training models to predict inputs/outputs given code and test cases entirely in natural language as Chain-of-Thought (CoT) rationales, we expose them to universal reasoning primitives -- like logic flow planning, state-space searching, decision tree traversal, and modular decomposition -- while decoupling structured reasoning from code-specific syntax and preserving procedural rigor. Experimental results demonstrate CodeI/O leads to consistent improvements across symbolic, scientific, logic, math & numerical, and commonsense reasoning tasks. By matching the existing ground-truth outputs or re-executing the code with predicted inputs, we can verify each prediction and further enhance the CoTs through multi-turn revision, resulting in CodeI/O++ and achieving higher performance. Our data and models are available at https://github.com/hkust-nlp/CodeIO.

Recent advancements in large language models (LLMs) have demonstrated significant progress in math and code reasoning capabilities. However, existing code benchmark are limited in their ability to evaluate the full spectrum of these capabilities, particularly at the competitive level. To bridge this gap, we introduce OJBench, a novel and challenging benchmark designed to assess the competitive-level code reasoning abilities of LLMs. OJBench comprises 232 programming competition problems from NOI and ICPC, providing a more rigorous test of models' reasoning skills. We conducted a comprehensive evaluation using OJBench on 37 models, including both closed-source and open-source models, reasoning-oriented and non-reasoning-oriented models. Our results indicate that even state-of-the-art reasoning-oriented models, such as o4-mini and Gemini-2.5-pro-exp, struggle with highly challenging competition-level problems. This highlights the significant challenges that models face in competitive-level code reasoning.

Code data has been shown to enhance the reasoning capabilities of large language models (LLMs), but it remains unclear which aspects of code are most responsible. We investigate this question with a systematic, data-centric framework. We construct parallel instruction datasets in ten programming languages and apply controlled perturbations that selectively disrupt structural or semantic properties of code. We then finetune LLMs from five model families and eight scales on each variant and evaluate their performance on natural language, math, and code tasks. Across 3,331 experiments, our results show that LLMs are more vulnerable to structural perturbations than semantic ones, particularly on math and code tasks. Appropriate abstractions like pseudocode and flowcharts can be as effective as code, while encoding the same information with fewer tokens without adhering to original syntax can often retain or even improve performance. Remarkably, even corrupted code with misleading signals remains competitive when surface-level regularities persist. Finally, syntactic styles also shape task-specific gains with Python favoring natural language reasoning and lower-level languages such as Java and Rust favoring math. Through our systematic framework, we aim to provide insight into how different properties of code influence reasoning and inform the design of training data for enhancing LLM reasoning capabilities.

Recent advances in Large Language Models (LLMs) and Vision Language Models (VLMs) have shown significant progress in mathematical reasoning, yet they still face a critical bottleneck with problems requiring visual assistance, such as drawing auxiliary lines or plotting functions to solve the problems. Most LLMs and VLMs are constrained to text-only reasoning chains, while multimodal unified models that can generate interleaved text and images lack the necessary precision and controllability for such tasks. To address this, we propose CodePlot-CoT, a code-driven Chain-of-Thought paradigm for "thinking with images" in mathematics. Our approach leverages the VLM to generate text reasoning as well as executable plotting code, which is then rendered into images as "visual thought", to solve mathematical problems. To achieve this, we first construct Math-VR, the first large-scale, bilingual dataset and benchmark for Mathematics problems with Visual Reasoning, comprising 178K samples. Second, to create high-quality training data, we develop a state-of-the-art image-to-code converter specialized for parsing complex mathematical figures into codes. Finally, using these training data, we train the CodePlot-CoT model for solving mathematical problems. Experimental results show that our model achieves up to 21% increase over base model on our new benchmark, fully validating the efficacy of our proposed code-driven reasoning paradigm. Our work opens a new direction for multimodal mathematical reasoning and provides the community with the first large-scale dataset, comprehensive benchmark, and strong approach for such problems. To facilitate future research, we make our datasets, code, and pretrained models publicly available at https://github.com/HKU-MMLab/Math-VR-CodePlot-CoT.

As AI systems progress, we rely more on them to make decisions with us and for us. To ensure that such decisions are aligned with human values, it is imperative for us to understand not only what decisions they make but also how they come to those decisions. Reasoning language models, which provide both final responses and (partially transparent) intermediate thinking traces, present a timely opportunity to study AI procedural reasoning. Unlike math and code problems which often have objectively correct answers, moral dilemmas are an excellent testbed for process-focused evaluation because they allow for multiple defensible conclusions. To do so, we present MoReBench: 1,000 moral scenarios, each paired with a set of rubric criteria that experts consider essential to include (or avoid) when reasoning about the scenarios. MoReBench contains over 23 thousand criteria including identifying moral considerations, weighing trade-offs, and giving actionable recommendations to cover cases on AI advising humans moral decisions as well as making moral decisions autonomously. Separately, we curate MoReBench-Theory: 150 examples to test whether AI can reason under five major frameworks in normative ethics. Our results show that scaling laws and existing benchmarks on math, code, and scientific reasoning tasks fail to predict models' abilities to perform moral reasoning. Models also show partiality towards specific moral frameworks (e.g., Benthamite Act Utilitarianism and Kantian Deontology), which might be side effects of popular training paradigms. Together, these benchmarks advance process-focused reasoning evaluation towards safer and more transparent AI.

Reinforcement learning (RL) has become a trending paradigm for training large language models (LLMs), particularly for reasoning tasks. Effective RL for LLMs requires massive parallelization and poses an urgent need for efficient training systems. Most existing large-scale RL systems for LLMs are synchronous by alternating generation and training in a batch setting, where the rollouts in each training batch are generated by the same (or latest) model. This stabilizes RL training but suffers from severe system-level inefficiency. Generation must wait until the longest output in the batch is completed before model update, resulting in GPU underutilization. We present AReaL, a fully asynchronous RL system that completely decouples generation from training. Rollout workers in AReaL continuously generate new outputs without waiting, while training workers update the model whenever a batch of data is collected. AReaL also incorporates a collection of system-level optimizations, leading to substantially higher GPU utilization. To stabilize RL training, AReaL balances the workload of rollout and training workers to control data staleness, and adopts a staleness-enhanced PPO variant to better handle outdated training samples. Extensive experiments on math and code reasoning benchmarks show that AReaL achieves up to 2.57times training speedup compared to the best synchronous systems with the same number of GPUs and matched or even improved final performance. The code of AReaL is available at https://github.com/inclusionAI/AReaL/.

Reinforcement learning (RL) with tree search has demonstrated superior performance in traditional reasoning tasks. Compared to conventional independent chain sampling strategies with outcome supervision, tree search enables better exploration of the reasoning space and provides dense, on-policy process rewards during RL training but remains under-explored in On-Policy LLM RL. We propose TreeRL, a reinforcement learning framework that directly incorporates on-policy tree search for RL training. Our approach includes intermediate supervision and eliminates the need for a separate reward model training. Existing approaches typically train a separate process reward model, which can suffer from distribution mismatch and reward hacking. We also introduce a cost-effective tree search approach that achieves higher search efficiency under the same generation token budget by strategically branching from high-uncertainty intermediate steps rather than using random branching. Experiments on challenging math and code reasoning benchmarks demonstrate that TreeRL achieves superior performance compared to traditional ChainRL, highlighting the potential of tree search for LLM. TreeRL is open-sourced at https://github.com/THUDM/TreeRL.

We introduce the Diffusion Chain of Lateral Thought (DCoLT), a reasoning framework for diffusion language models. DCoLT treats each intermediate step in the reverse diffusion process as a latent "thinking" action and optimizes the entire reasoning trajectory to maximize the reward on the correctness of the final answer with outcome-based Reinforcement Learning (RL). Unlike traditional Chain-of-Thought (CoT) methods that follow a causal, linear thinking process, DCoLT allows bidirectional, non-linear reasoning with no strict rule on grammatical correctness amid its intermediate steps of thought. We implement DCoLT on two representative Diffusion Language Models (DLMs). First, we choose SEDD as a representative continuous-time discrete diffusion model, where its concrete score derives a probabilistic policy to maximize the RL reward over the entire sequence of intermediate diffusion steps. We further consider the discrete-time masked diffusion language model -- LLaDA, and find that the order to predict and unmask tokens plays an essential role to optimize its RL action resulting from the ranking-based Unmasking Policy Module (UPM) defined by the Plackett-Luce model. Experiments on both math and code generation tasks show that using only public data and 16 H800 GPUs, DCoLT-reinforced DLMs outperform other DLMs trained by SFT or RL or even both. Notably, DCoLT-reinforced LLaDA boosts its reasoning accuracy by +9.8%, +5.7%, +11.4%, +19.5% on GSM8K, MATH, MBPP, and HumanEval.

We propose FlowRL: matching the full reward distribution via flow balancing instead of maximizing rewards in large language model (LLM) reinforcement learning (RL). Recent advanced reasoning models adopt reward-maximizing methods (\eg, PPO and GRPO), which tend to over-optimize dominant reward signals while neglecting less frequent but valid reasoning paths, thus reducing diversity. In contrast, we transform scalar rewards into a normalized target distribution using a learnable partition function, and then minimize the reverse KL divergence between the policy and the target distribution. We implement this idea as a flow-balanced optimization method that promotes diverse exploration and generalizable reasoning trajectories. We conduct experiments on math and code reasoning tasks: FlowRL achieves a significant average improvement of 10.0% over GRPO and 5.1% over PPO on math benchmarks, and performs consistently better on code reasoning tasks. These results highlight reward distribution-matching as a key step toward efficient exploration and diverse reasoning in LLM reinforcement learning.

We introduce LogQuant, a groundbreaking 2-bit quantization technique for KV Cache in large language model (LLM) inference, delivering substantial memory savings while preserving superior performance. Previous methods either assume that later tokens are more important or attempt to predict important tokens based on earlier attention patterns. Both approaches, however, can result in performance bottlenecks or frequent mispredictions. LogQuant takes a different approach. By applying a log-based filtering mechanism, it selectively compresses the KV Cache across the entire context, achieving better performance with the same or even reduced memory footprint compared to existing methods. In benchmark tests, it enhances throughput by 25% and boosts batch size by 60% without increasing memory consumption. For challenging tasks such as Math and Code Completion, LogQuant improves accuracy by 40% to 200% at the same compression ratio, outperforming comparable techniques.LogQuant integrates effortlessly with popular inference frameworks like Python's transformers library. Implementation can be available in https://github.com/Concyclics/LogQuantKV.

It is well-known that a diverse corpus is critical for training large language models, which are typically constructed from a mixture of various domains. In general, previous efforts resort to sampling training data from different domains with static proportions, as well as adjusting data proportions during training. However, few methods have addressed the complexities of domain-adaptive continual pre-training. To fill this gap, we propose Velocitune, a novel framework dynamically assesses learning velocity and adjusts data proportions accordingly, favoring slower-learning domains while shunning faster-learning ones, which is guided by a scaling law to indicate the desired learning goal for each domain with less associated cost. To evaluate the effectiveness of Velocitune, we conduct experiments in a reasoning-focused dataset with CodeLlama, as well as in a corpus specialised for system command generation with Llama3 and Mistral. Velocitune achieves performance gains in both math and code reasoning tasks and command-line generation benchmarks. Further analysis reveals that key factors driving Velocitune's effectiveness include target loss prediction and data ordering.

Direct Preference Optimization (DPO) has proven effective in complex reasoning tasks like math word problems and code generation. However, when applied to Text-to-SQL datasets, it often fails to improve performance and can even degrade it. Our investigation reveals the root cause: unlike math and code tasks, which naturally integrate Chain-of-Thought (CoT) reasoning with DPO, Text-to-SQL datasets typically include only final answers (gold SQL queries) without detailed CoT solutions. By augmenting Text-to-SQL datasets with synthetic CoT solutions, we achieve, for the first time, consistent and significant performance improvements using DPO. Our analysis shows that CoT reasoning is crucial for unlocking DPO's potential, as it mitigates reward hacking, strengthens discriminative capabilities, and improves scalability. These findings offer valuable insights for building more robust Text-to-SQL models. To support further research, we publicly release the code and CoT-enhanced datasets.

Can scaling transform reasoning? In this work, we explore the untapped potential of scaling Long Chain-of-Thought (Long-CoT) data to 1000k samples, pioneering the development of a slow-thinking model, RedStar. Through extensive experiments with various LLMs and different sizes, we uncover the ingredients for specialization and scale for Long-CoT training. Surprisingly, even smaller models show significant performance gains with limited data, revealing the sample efficiency of Long-CoT and the critical role of sample difficulty in the learning process. Our findings demonstrate that Long-CoT reasoning can be effectively triggered with just a few thousand examples, while larger models achieve unparalleled improvements. We also introduce reinforcement learning (RL)-scale training as a promising direction for advancing slow-thinking systems. RedStar shines across domains: on the MATH-Hard benchmark, RedStar-code-math boosts performance from 66.2\% to 81.6\%, and on the USA Math Olympiad (AIME), it solves 46.7\% of problems using only 21k mixed-code-math datasets. In multimodal tasks like GeoQA and MathVista-GEO, RedStar-Geo achieves competitive results with minimal Long-CoT data, outperforming other slow-thinking systems like QvQ-Preview. Compared to QwQ, RedStar strikes the perfect balance between reasoning and generalizability. Our work highlights that, with careful tuning, scaling Long-CoT can unlock extraordinary reasoning capabilities-even with limited dataset and set a new standard for slow-thinking models across diverse challenges. Our data and models are released at https://huggingface.co/RedStar-Reasoning.

The BigCode project, an open-scientific collaboration focused on the responsible development of Large Language Models for Code (Code LLMs), introduces StarCoder2. In partnership with Software Heritage (SWH), we build The Stack v2 on top of the digital commons of their source code archive. Alongside the SWH repositories spanning 619 programming languages, we carefully select other high-quality data sources, such as GitHub pull requests, Kaggle notebooks, and code documentation. This results in a training set that is 4x larger than the first StarCoder dataset. We train StarCoder2 models with 3B, 7B, and 15B parameters on 3.3 to 4.3 trillion tokens and thoroughly evaluate them on a comprehensive set of Code LLM benchmarks. We find that our small model, StarCoder2-3B, outperforms other Code LLMs of similar size on most benchmarks, and also outperforms StarCoderBase-15B. Our large model, StarCoder2- 15B, significantly outperforms other models of comparable size. In addition, it matches or outperforms CodeLlama-34B, a model more than twice its size. Although DeepSeekCoder- 33B is the best-performing model at code completion for high-resource languages, we find that StarCoder2-15B outperforms it on math and code reasoning benchmarks, as well as several low-resource languages. We make the model weights available under an OpenRAIL license and ensure full transparency regarding the training data by releasing the SoftWare Heritage persistent IDentifiers (SWHIDs) of the source code data.

Since the release of ChatGPT, large language models (LLMs) have demonstrated remarkable capabilities across various domains. A key challenge in developing these general capabilities is efficiently sourcing diverse, high-quality data. This becomes especially critical in reasoning-related tasks with sandbox checkers, such as math or code, where the goal is to generate correct solutions to specific problems with higher probability. In this work, we introduce Flaming-hot Initiation with Regular Execution (FIRE) sampling, a simple yet highly effective method to efficiently find good responses. Our empirical findings show that FIRE sampling enhances inference-time generation quality and also benefits training in the alignment stage. Furthermore, we explore how FIRE sampling improves performance by promoting diversity and analyze the impact of employing FIRE at different positions within a response.

Reasoning methods that adaptively allocate test-time compute have advanced LLM performance on easy to verify domains such as math and code. In this work, we study how to utilize this approach to train models that exhibit a degree of robustness to safety vulnerabilities, and show that doing so can provide benefits. We build a recipe called TARS (Training Adaptive Reasoners for Safety), a reinforcement learning (RL) approach that trains models to reason about safety using chain-of-thought traces and a reward signal that balances safety with task completion. To build TARS, we identify three critical design choices: (1) a "lightweight" warmstart SFT stage, (2) a mix of harmful, harmless, and ambiguous prompts to prevent shortcut behaviors such as too many refusals, and (3) a reward function to prevent degeneration of reasoning capabilities during training. Models trained with TARS exhibit adaptive behaviors by spending more compute on ambiguous queries, leading to better safety-refusal trade-offs. They also internally learn to better distinguish between safe and unsafe prompts and attain greater robustness to both white-box (e.g., GCG) and black-box attacks (e.g., PAIR). Overall, our work provides an effective, open recipe for training LLMs against jailbreaks and harmful requests by reasoning per prompt.

Despite significant advancements in the general capability of large language models (LLMs), they continue to struggle with consistent and accurate reasoning, especially in complex tasks such as mathematical and code reasoning. One key limitation is that LLMs are trained primarily on correct solutions, reducing their ability to detect and learn from errors, which hampers their ability to reliably verify and rank outputs. To address this, we scale up the inference-time computation by generating multiple reasoning paths and employing verifiers to assess and rank the generated outputs by correctness. To facilitate this, we introduce a comprehensive dataset consisting of correct and incorrect solutions for math and code tasks, generated by multiple LLMs. This diverse set of solutions enables verifiers to more effectively distinguish and rank correct answers from erroneous outputs. The training methods for building verifiers were selected based on an extensive comparison of existing approaches. Moreover, to leverage the unique strengths of different reasoning strategies, we propose a novel collaborative method integrating Chain-of-Thought (CoT) and Program-of-Thought (PoT) solutions for verification. CoT provides a clear, step-by-step reasoning process that enhances interpretability, while PoT, being executable, offers a precise and error-sensitive validation mechanism. By taking both of their strengths, our approach significantly improves the accuracy and reliability of reasoning verification. Our verifiers, Math-Rev and Code-Rev, demonstrate substantial performance gains to existing LLMs, achieving state-of-the-art results on benchmarks such as GSM8k and MATH and even outperforming GPT-4o with Qwen-72B-Instruct as the reasoner.

Recent advancements in large language models (LLMs) have shifted focus toward scaling inference-time compute, improving performance without retraining the model. A common approach is to sample multiple outputs in parallel, and select one of these as the final output. However, work to date has focused on English and a handful of domains such as math and code. In contrast, we are most interested in techniques that generalize across open-ended tasks, formally verifiable tasks, and across languages. In this work, we study how to robustly scale inference-time compute for open-ended generative tasks in a multilingual, multi-task setting. Our findings show that both sampling strategy based on temperature variation and selection strategy must be adapted to account for diverse domains and varied language settings. We evaluate existing selection methods, revealing that strategies effective in English often fail to generalize across languages. We propose novel sampling and selection strategies specifically adapted for multilingual and multi-task inference scenarios, and show they yield notable gains across languages and tasks. In particular, our combined sampling and selection methods lead to an average +6.8 jump in win-rates for our 8B models on m-ArenaHard-v2.0 prompts, against proprietary models such as Gemini. At larger scale, Command-A (111B model) equipped with our methods, shows +9.0 improvement in win-rates on the same benchmark with just five samples against single-sample decoding, a substantial increase at minimal cost. Our results underscore the need for language- and task-aware approaches to inference-time compute, aiming to democratize performance improvements in underrepresented languages.

Reinforcement learning (RL) has emerged as a promising approach to improve large language model (LLM) reasoning, yet most open efforts focus narrowly on math and code, limiting our understanding of its broader applicability to general reasoning. A key challenge lies in the lack of reliable, scalable RL reward signals across diverse reasoning domains. We introduce Guru, a curated RL reasoning corpus of 92K verifiable examples spanning six reasoning domains--Math, Code, Science, Logic, Simulation, and Tabular--each built through domain-specific reward design, deduplication, and filtering to ensure reliability and effectiveness for RL training. Based on Guru, we systematically revisit established findings in RL for LLM reasoning and observe significant variation across domains. For example, while prior work suggests that RL primarily elicits existing knowledge from pretrained models, our results reveal a more nuanced pattern: domains frequently seen during pretraining (Math, Code, Science) easily benefit from cross-domain RL training, while domains with limited pretraining exposure (Logic, Simulation, and Tabular) require in-domain training to achieve meaningful performance gains, suggesting that RL is likely to facilitate genuine skill acquisition. Finally, we present Guru-7B and Guru-32B, two models that achieve state-of-the-art performance among open models RL-trained with publicly available data, outperforming best baselines by 7.9% and 6.7% on our 17-task evaluation suite across six reasoning domains. We also show that our models effectively improve the Pass@k performance of their base models, particularly on complex tasks less likely to appear in pretraining data. We release data, models, training and evaluation code to facilitate general-purpose reasoning at: https://github.com/LLM360/Reasoning360

We introduce CCI4.0, a large-scale bilingual pre-training dataset engineered for superior data quality and diverse human-like reasoning trajectory. CCI4.0 occupies roughly 35 TB of disk space and comprises two sub-datasets: CCI4.0-M2-Base and CCI4.0-M2-CoT. CCI4.0-M2-Base combines a 5.2 TB carefully curated Chinese web corpus, a 22.5 TB English subset from Nemotron-CC, and diverse sources from math, wiki, arxiv, and code. Although these data are mostly sourced from well-processed datasets, the quality standards of various domains are dynamic and require extensive expert experience and labor to process. So, we propose a novel pipeline justifying data quality mainly based on models through two-stage deduplication, multiclassifier quality scoring, and domain-aware fluency filtering. We extract 4.5 billion pieces of CoT(Chain-of-Thought) templates, named CCI4.0-M2-CoT. Differing from the distillation of CoT from larger models, our proposed staged CoT extraction exemplifies diverse reasoning patterns and significantly decreases the possibility of hallucination. Empirical evaluations demonstrate that LLMs pre-trained in CCI4.0 benefit from cleaner, more reliable training signals, yielding consistent improvements in downstream tasks, especially in math and code reflection tasks. Our results underscore the critical role of rigorous data curation and human thinking templates in advancing LLM performance, shedding some light on automatically processing pretraining corpora.

Sparse autoencoders have recently produced dictionaries of high-dimensional vectors corresponding to the universe of concepts represented by large language models. We find that this concept universe has interesting structure at three levels: 1) The "atomic" small-scale structure contains "crystals" whose faces are parallelograms or trapezoids, generalizing well-known examples such as (man-woman-king-queen). We find that the quality of such parallelograms and associated function vectors improves greatly when projecting out global distractor directions such as word length, which is efficiently done with linear discriminant analysis. 2) The "brain" intermediate-scale structure has significant spatial modularity; for example, math and code features form a "lobe" akin to functional lobes seen in neural fMRI images. We quantify the spatial locality of these lobes with multiple metrics and find that clusters of co-occurring features, at coarse enough scale, also cluster together spatially far more than one would expect if feature geometry were random. 3) The "galaxy" scale large-scale structure of the feature point cloud is not isotropic, but instead has a power law of eigenvalues with steepest slope in middle layers. We also quantify how the clustering entropy depends on the layer.

Large language models(LLMs) with powerful language generation and reasoning capabilities have already achieved success in many domains, e.g., math and code generation. However, due to the lacking of physical world's corpus and knowledge during training, they usually fail to solve many real-life tasks in the urban space. In this paper, we propose CityGPT, a systematic framework for enhancing the capability of LLMs on understanding urban space and solving the related urban tasks by building a city-scale world model in the model. First, we construct a diverse instruction tuning dataset CityInstruction for injecting urban knowledge and enhancing spatial reasoning capability effectively. By using a mixture of CityInstruction and general instruction data, we fine-tune various LLMs (e.g., ChatGLM3-6B, Qwen1.5 and LLama3 series) to enhance their capability without sacrificing general abilities. To further validate the effectiveness of proposed methods, we construct a comprehensive benchmark CityEval to evaluate the capability of LLMs on diverse urban scenarios and problems. Extensive evaluation results demonstrate that small LLMs trained with CityInstruction can achieve competitive performance with commercial LLMs in the comprehensive evaluation of CityEval. The source codes are openly accessible to the research community via https://github.com/tsinghua-fib-lab/CityGPT.

Continual Pre-Training (CPT) on Large Language Models (LLMs) has been widely used to expand the model's fundamental understanding of specific downstream domains (e.g., math and code). For the CPT on domain-specific LLMs, one important question is how to choose the optimal mixture ratio between the general-corpus (e.g., Dolma, Slim-pajama) and the downstream domain-corpus. Existing methods usually adopt laborious human efforts by grid-searching on a set of mixture ratios, which require high GPU training consumption costs. Besides, we cannot guarantee the selected ratio is optimal for the specific domain. To address the limitations of existing methods, inspired by the Scaling Law for performance prediction, we propose to investigate the Scaling Law of the Domain-specific Continual Pre-Training (D-CPT Law) to decide the optimal mixture ratio with acceptable training costs for LLMs of different sizes. Specifically, by fitting the D-CPT Law, we can easily predict the general and downstream performance of arbitrary mixture ratios, model sizes, and dataset sizes using small-scale training costs on limited experiments. Moreover, we also extend our standard D-CPT Law on cross-domain settings and propose the Cross-Domain D-CPT Law to predict the D-CPT law of target domains, where very small training costs (about 1% of the normal training costs) are needed for the target domains. Comprehensive experimental results on six downstream domains demonstrate the effectiveness and generalizability of our proposed D-CPT Law and Cross-Domain D-CPT Law.

The success of ChatGPT has recently attracted numerous efforts to replicate it, with instruction-tuning strategies being a key factor in achieving remarkable results. Instruction-tuning not only significantly enhances the model's performance and generalization but also makes the model's generated results more consistent with human speech patterns. However current research rarely studies the impact of different amounts of instruction data on model performance, especially in the real-world use cases. In this paper we explore the performance of large language models based on instruction tuning across different scales of instruction data. An evaluation dataset consisting of 12 major online use cases is constructed in the experiment. With Bloomz-7B1-mt as the base model, the results show that 1) merely increasing the amount of instruction data leads to continuous improvement in tasks such as open-ended generation, 2) in tasks such as math and code, the model performance curve remains quite flat while increasing data size. We further analyze the possible causes of these phenomena and propose potential future research directions such as effectively selecting high-quality training data, scaling base models and training methods specialized for hard tasks. We will release our training and evaluation datasets, as well as model checkpoints.

As large language models (LLMs) continue to advance, the need for up-to-date and well-organized benchmarks becomes increasingly critical. However, many existing datasets are scattered, difficult to manage, and make it challenging to perform evaluations tailored to specific needs or domains, despite the growing importance of domain-specific models in areas such as math or code. In this paper, we introduce BenchHub, a dynamic benchmark repository that empowers researchers and developers to evaluate LLMs more effectively. BenchHub aggregates and automatically classifies benchmark datasets from diverse domains, integrating 303K questions across 38 benchmarks. It is designed to support continuous updates and scalable data management, enabling flexible and customizable evaluation tailored to various domains or use cases. Through extensive experiments with various LLM families, we demonstrate that model performance varies significantly across domain-specific subsets, emphasizing the importance of domain-aware benchmarking. We believe BenchHub can encourage better dataset reuse, more transparent model comparisons, and easier identification of underrepresented areas in existing benchmarks, offering a critical infrastructure for advancing LLM evaluation research.

As language model (LM) outputs get more and more natural, it is becoming more difficult than ever to evaluate their quality. Simultaneously, increasing LMs' "thinking" time through scaling test-time compute has proven an effective technique to solve challenging problems in domains such as math and code. This raises a natural question: can an LM's evaluation capability also be improved by spending more test-time compute? To answer this, we investigate employing reasoning models-LMs that natively generate long chain-of-thought reasoning-as evaluators. Specifically, we examine methods to leverage more test-time compute by (1) using reasoning models, and (2) prompting these models to evaluate not only the response as a whole (i.e., outcome evaluation) but also assess each step in the response separately (i.e., process evaluation). In experiments, we observe that the evaluator's performance improves monotonically when generating more reasoning tokens, similar to the trends observed in LM-based generation. Furthermore, we use these more accurate evaluators to rerank multiple generations, and demonstrate that spending more compute at evaluation time can be as effective as using more compute at generation time in improving an LM's problem-solving capability.

Recent reasoning models through test-time scaling have demonstrated that long chain-of-thoughts can unlock substantial performance boosts in hard reasoning tasks such as math and code. However, the benefit of such long thoughts for system-2 reasoning is relatively less explored in other domains such as perceptual tasks where shallower, system-1 reasoning seems sufficient. In this paper, we introduce LongPerceptualThoughts, a new synthetic dataset with 30K long-thought traces for perceptual tasks. The key challenges in synthesizing elaborate reasoning thoughts for perceptual tasks are that off-the-shelf models are not yet equipped with such thinking behavior and that it is not straightforward to build a reliable process verifier for perceptual tasks. Thus, we propose a novel three-stage data synthesis framework that first synthesizes verifiable multiple-choice questions from dense image descriptions, then extracts simple CoTs from VLMs for those verifiable problems, and finally expands those simple thoughts to elaborate long thoughts via frontier reasoning models. In controlled experiments with a strong instruction-tuned 7B model, we demonstrate notable improvements over existing visual reasoning data-generation methods. Our model, trained on the generated dataset, achieves an average +3.4 points improvement over 5 vision-centric benchmarks, including +11.8 points on V^* Bench. Notably, despite being tuned for vision tasks, it also improves performance on the text reasoning benchmark, MMLU-Pro, by +2 points.

Recent advances in large language models (LLMs) have enabled automatic generation of chain-of-thought (CoT) reasoning, leading to strong performance on tasks such as math and code. However, when reasoning steps reflect social stereotypes (e.g., those related to gender, race or age), they can reinforce harmful associations and lead to misleading conclusions. We present the first systematic evaluation of social bias within LLM-generated reasoning, using the BBQ dataset to analyze both prediction accuracy and bias. Our study spans a wide range of mainstream reasoning models, including instruction-tuned and CoT-augmented variants of DeepSeek-R1 (8B/32B), ChatGPT, and other open-source LLMs. We quantify how biased reasoning steps correlate with incorrect predictions and often lead to stereotype expression. To mitigate reasoning-induced bias, we propose Answer Distribution as Bias Proxy (ADBP), a lightweight mitigation method that detects bias by tracking how model predictions change across incremental reasoning steps. ADBP outperforms a stereotype-free baseline in most cases, mitigating bias and improving the accuracy of LLM outputs. Code will be released upon paper acceptance.

Large language models (LLMs) have seen considerable advancements in natural language understanding tasks, yet there remains a gap to bridge before attaining true artificial general intelligence, especially concerning shortcomings in mathematical reasoning capabilities. We postulate that the inherent nature of LLM training, which focuses on predicting probabilities of next token, presents challenges in effectively modeling mathematical reasoning that demands exact calculations, both from data-driven and theoretical standpoints. In this paper, we address this challenge by enriching the data landscape and introducing a novel math dataset, enhanced with a capability to utilize a Python code interpreter. This dataset is derived from GSM8K and MATH and has been further refined through a combination of GPT-4 annotations, human review, and self-training processes, where the errors in the original GSM8K training set have been fixed. Additionally, we propose a tentative, easily replicable protocol for the fine-tuning of math-specific LLMs, which has led to a significant improvement in the performance of a 7B-parameter LLM on the GSM8K and MATH datasets. We are committed to advancing the field of mathematical reasoning in LLMs and, to that end, we have made the model checkpoints and will make the dataset publicly available. We hope this will facilitate further research and development within the community.

Underlying data distributions of natural language, programming code, and mathematical symbols vary vastly, presenting a complex challenge for large language models (LLMs) that strive to achieve high performance across all three domains simultaneously. Achieving a very high level of proficiency for an LLM within a specific domain often requires extensive training with relevant corpora, which is typically accompanied by a sacrifice in performance in other domains. In this paper, we propose to fuse models that are already highly-specialized directly. The proposed fusing framework, UltraFuser, consists of three distinct specialists that are already sufficiently trained on language, coding, and mathematics. A token-level gating mechanism is introduced to blend the specialists' outputs. A two-stage training strategy accompanied by balanced sampling is designed to ensure stability. To effectively train the fused model, we further construct a high-quality supervised instruction tuning dataset, UltraChat 2, which includes text, code, and mathematical content. This dataset comprises approximately 300,000 instructions and covers a wide range of topics in each domain. Experiments show that our model could simultaneously achieve mastery of the three crucial domains.

The recently released GPT-4 Code Interpreter has demonstrated remarkable proficiency in solving challenging math problems, primarily attributed to its ability to seamlessly reason with natural language, generate code, execute code, and continue reasoning based on the execution output. In this paper, we present a method to fine-tune open-source language models, enabling them to use code for modeling and deriving math equations and, consequently, enhancing their mathematical reasoning abilities. We propose a method of generating novel and high-quality datasets with math problems and their code-based solutions, referred to as MathCodeInstruct. Each solution interleaves natural language, code, and execution results. We also introduce a customized supervised fine-tuning and inference approach. This approach yields the MathCoder models, a family of models capable of generating code-based solutions for solving challenging math problems. Impressively, the MathCoder models achieve state-of-the-art scores among open-source LLMs on the MATH (45.2%) and GSM8K (83.9%) datasets, substantially outperforming other open-source alternatives. Notably, the MathCoder model not only surpasses ChatGPT-3.5 and PaLM-2 on GSM8K and MATH but also outperforms GPT-4 on the competition-level MATH dataset. The dataset and models will be released at https://github.com/mathllm/MathCoder.

Humans generally acquire new skills without compromising the old; however, the opposite holds for Large Language Models (LLMs), e.g., from LLaMA to CodeLLaMA. To this end, we propose a new post-pretraining method for LLMs with an expansion of Transformer blocks. We tune the expanded blocks using only new corpus, efficiently and effectively improving the model's knowledge without catastrophic forgetting. In this paper, we experiment on the corpus of code and math, yielding LLaMA Pro-8.3B, a versatile foundation model initialized from LLaMA2-7B, excelling in general tasks, programming, and mathematics. LLaMA Pro and its instruction-following counterpart (LLaMA Pro-Instruct) achieve advanced performance among various benchmarks, demonstrating superiority over existing open models in the LLaMA family and the immense potential of reasoning and addressing diverse tasks as an intelligent agent. Our findings provide valuable insights into integrating natural and programming languages, laying a solid foundation for developing advanced language agents that operate effectively in various environments.

Large language models (LLMs) demonstrate remarkable reasoning capabilities in tasks such as algorithmic coding and mathematical problem-solving. Recent methods have improved reasoning through expanded corpus and multistage training combining reinforcement learning and supervised fine-tuning. Although some methods suggest that small but targeted dataset can incentivize reasoning via only distillation, a reasoning scaling laws is still taking shape, increasing computational costs. To address this, we propose a data-efficient distillation framework (DED) that optimizes the Pareto frontier of reasoning distillation. Inspired by the on-policy learning and diverse roll-out strategies of reinforcement learning, the key idea of our approach is threefold: (1) We identify that benchmark scores alone do not determine an effective teacher model. Through comprehensive comparisons of leading reasoning LLMs, we develop a method to select an optimal teacher model. (2) While scaling distillation can enhance reasoning, it often degrades out-of-domain performance. A carefully curated, smaller corpus achieves a balanced trade-off between in-domain and out-of-domain capabilities. (3) Diverse reasoning trajectories encourage the student model to develop robust reasoning skills. We validate our method through evaluations on mathematical reasoning (AIME 2024/2025, MATH-500) and code generation (LiveCodeBench), achieving state-of-the-art results with only 0.8k carefully curated examples, bypassing the need for extensive scaling. Our systematic analysis demonstrates that DED outperforms existing methods by considering factors beyond superficial hardness, token length, or teacher model capability. This work offers a practical and efficient pathway to advanced reasoning while preserving general capabilities.

Reinforcement learning (RL) has shown great effectiveness for fine-tuning large language models (LLMs) using tasks that are challenging yet easily verifiable, such as math reasoning or code generation. However, extending this success to visual perception in vision-language models (VLMs) has been impeded by the scarcity of vision-centric tasks that are simultaneously challenging and unambiguously verifiable. To this end, we introduce ViCrit (Visual Caption Hallucination Critic), an RL proxy task that trains VLMs to localize a subtle, synthetic visual hallucination injected into paragraphs of human-written image captions. Starting from a 200-word captions, we inject a single, subtle visual description error-altering a few words on objects, attributes, counts, or spatial relations-and task the model to pinpoint the corrupted span given the image and the modified caption. This formulation preserves the full perceptual difficulty while providing a binary, exact-match reward that is easy to compute and unambiguous. Models trained with the ViCrit Task exhibit substantial gains across a variety of VL benchmarks. Crucially, the improvements transfer beyond natural-image training data to abstract image reasoning and visual math, showing promises of learning to perceive rather than barely memorizing seen objects. To facilitate evaluation, we further introduce ViCrit-Bench, a category-balanced diagnostic benchmark that systematically probes perception errors across diverse image domains and error types. Together, our results demonstrate that fine-grained hallucination criticism is an effective and generalizable objective for enhancing visual perception in VLMs.

Finetuning large language models on instruction data is crucial for enhancing pre-trained knowledge and improving instruction-following capabilities. As instruction datasets proliferate, selecting optimal data for effective training becomes increasingly important. This work addresses the question: How can we determine the optimal subset of data for effective training? While existing research often emphasizes local criteria like instance quality for subset selection, we argue that a global approach focused on data diversity is more critical. Our method employs k-means clustering to ensure the selected subset effectively represents the full dataset. We propose an iterative refinement method inspired by active learning techniques to resample instances from clusters, reassessing each cluster's importance and sampling weight in every training iteration. This approach reduces the effect of outliers and automatically filters out clusters containing low-quality data. Through extensive evaluation across natural language reasoning, general world knowledge, code and math reasoning tasks, and by fine-tuning models from various families, we observe consistent improvements, achieving a 7% increase over random selection and a 3.8% improvement over state-of-the-art sampling methods. Our work highlights the significance of diversity-first sampling when finetuning LLMs to enhance performance across a broad array of evaluation tasks. Our code is available at https://github.com/for-ai/iterative-data-selection.

While foundation models update slowly due to resource-intensive training requirements, domain-specific models evolve between updates. Model merging aims to combine multiple expert models into a single, more capable model, thereby reducing storage and serving costs while supporting decentralized model development. Despite its potential, previous studies have primarily focused on merging visual classification models or Large Language Models (LLMs) for code and math tasks. Multimodal Large Language Models (MLLMs), which extend the capabilities of LLMs through large-scale multimodal training, have gained traction. However, there lacks a benchmark for model merging research that clearly divides the tasks for MLLM training and evaluation. In this paper, (i) we introduce the model merging benchmark for MLLMs, which includes multiple tasks such as VQA, Geometry, Chart, OCR, and Grounding, providing both LoRA and full fine-tuning models. Moreover, we explore how model merging can combine different modalities (e.g., vision-language, audio-language, and video-language models), moving toward the Omni-language model. (ii) We implement 10 model merging algorithms on the benchmark. Furthermore, we propose a novel method that removes noise from task vectors and robustly optimizes the merged vector based on a loss defined over task vector interactions, achieving an average performance gain of 2.48%. (iii) We find that model merging offers a promising way for building improved MLLMs without requiring data training. Our results also demonstrate that the complementarity among multiple modalities outperforms individual modalities.

Linear Recurrent Neural Networks (LRNNs) such as Mamba, RWKV, GLA, mLSTM, and DeltaNet have emerged as efficient alternatives to Transformers for long sequences. However, both Transformers and LRNNs struggle to perform state-tracking, which may impair performance in tasks such as code evaluation. In one forward pass, current architectures are unable to solve even parity, the simplest state-tracking task, which non-linear RNNs can handle effectively. Recently, Sarrof et al. (2024) demonstrated that the failure of LRNNs like Mamba to solve parity stems from restricting the value range of their diagonal state-transition matrices to [0, 1] and that incorporating negative values can resolve this issue. We extend this result to non-diagonal LRNNs such as DeltaNet. We prove that finite precision LRNNs with state-transition matrices having only positive eigenvalues cannot solve parity, while non-triangular matrices are needed to count modulo 3. Notably, we also prove that LRNNs can learn any regular language when their state-transition matrices are products of identity minus vector outer product matrices, each with eigenvalues in the range [-1, 1]. Our experiments confirm that extending the eigenvalue range of Mamba and DeltaNet to include negative values not only enables them to solve parity but consistently improves their performance on state-tracking tasks. We also show that state-tracking enabled LRNNs can be pretrained stably and efficiently at scale (1.3B parameters), achieving competitive performance on language modeling and showing promise on code and math tasks.

Code has been shown to be effective in enhancing the mathematical reasoning abilities of large language models due to its precision and accuracy. Previous works involving continued mathematical pretraining often include code that utilizes math-related packages, which are primarily designed for fields such as engineering, machine learning, signal processing, or module testing, rather than being directly focused on mathematical reasoning. In this paper, we introduce a novel method for generating mathematical code accompanied with corresponding reasoning steps for continued pretraining. Our approach begins with the construction of a high-quality mathematical continued pretraining dataset by incorporating math-related web data, code using mathematical packages, math textbooks, and synthetic data. Next, we construct reasoning steps by extracting LaTeX expressions, the conditions needed for the expressions, and the results of the expressions from the previously collected dataset. Based on this extracted information, we generate corresponding code to accurately capture the mathematical reasoning process. Appending the generated code to each reasoning step results in data consisting of paired natural language reasoning steps and their corresponding code. Combining this data with the original dataset results in a 19.2B-token high-performing mathematical pretraining corpus, which we name MathCode-Pile. Training several popular base models with this corpus significantly improves their mathematical abilities, leading to the creation of the MathCoder2 family of models. All of our data processing and training code is open-sourced, ensuring full transparency and easy reproducibility of the entire data collection and training pipeline. The code is released at https://github.com/mathllm/MathCoder2 .

Recent progress in large language models (LLMs) like GPT-4 and PaLM-2 has brought significant advancements in addressing math reasoning problems. In particular, OpenAI's latest version of GPT-4, known as GPT-4 Code Interpreter, shows remarkable performance on challenging math datasets. In this paper, we explore the effect of code on enhancing LLMs' reasoning capability by introducing different constraints on the Code Usage Frequency of GPT-4 Code Interpreter. We found that its success can be largely attributed to its powerful skills in generating and executing code, evaluating the output of code execution, and rectifying its solution when receiving unreasonable outputs. Based on this insight, we propose a novel and effective prompting method, explicit code-based self-verification~(CSV), to further boost the mathematical reasoning potential of GPT-4 Code Interpreter. This method employs a zero-shot prompt on GPT-4 Code Interpreter to encourage it to use code to self-verify its answers. In instances where the verification state registers as ``False'', the model shall automatically amend its solution, analogous to our approach of rectifying errors during a mathematics examination. Furthermore, we recognize that the states of the verification result indicate the confidence of a solution, which can improve the effectiveness of majority voting. With GPT-4 Code Interpreter and CSV, we achieve an impressive zero-shot accuracy on MATH dataset (53.9\% to 84.3\%).

Pre-training Large Language Models (LLMs) on high-quality, meticulously curated datasets is widely recognized as critical for enhancing their performance and generalization capabilities. This study explores the untapped potential of Common Crawl as a comprehensive and flexible resource for pre-training LLMs, addressing both general-purpose language understanding and specialized domain knowledge. We introduce RedStone, an innovative and scalable pipeline engineered to extract and process data from Common Crawl, facilitating the creation of extensive and varied pre-training datasets. Unlike traditional datasets, which often require expensive curation and domain-specific expertise, RedStone leverages the breadth of Common Crawl to deliver datasets tailored to a wide array of domains. In this work, we exemplify its capability by constructing pre-training datasets across multiple fields, including general language understanding, code, mathematics, and question-answering tasks. The flexibility of RedStone allows for easy adaptation to other specialized domains, significantly lowering the barrier to creating valuable domain-specific datasets. Our findings demonstrate that Common Crawl, when harnessed through effective pipelines like RedStone, can serve as a rich, renewable source of pre-training data, unlocking new avenues for domain adaptation and knowledge discovery in LLMs. This work also underscores the importance of innovative data acquisition strategies and highlights the role of web-scale data as a powerful resource in the continued evolution of LLMs. RedStone code and data samples will be publicly available at https://aka.ms/redstone.

Large language models (LLMs) have demonstrated impressive reasoning capabilities, particularly in textual mathematical problem-solving. However, existing open-source image instruction fine-tuning datasets, containing limited question-answer pairs per image, do not fully exploit visual information to enhance the multimodal mathematical reasoning capabilities of Multimodal LLMs (MLLMs). To bridge this gap, we address the lack of high-quality, diverse multimodal mathematical datasets by collecting 40K high-quality images with question-answer pairs from 24 existing datasets and synthesizing 320K new pairs, creating the MathV360K dataset, which enhances both the breadth and depth of multimodal mathematical questions. We introduce Math-LLaVA, a LLaVA-1.5-based model fine-tuned with MathV360K. This novel approach significantly improves the multimodal mathematical reasoning capabilities of LLaVA-1.5, achieving a 19-point increase and comparable performance to GPT-4V on MathVista's minitest split. Furthermore, Math-LLaVA demonstrates enhanced generalizability, showing substantial improvements on the MMMU benchmark. Our research highlights the importance of dataset diversity and synthesis in advancing MLLMs' mathematical reasoning abilities. The code and data are available at: https://github.com/HZQ950419/Math-LLaVA.

Multimodal Large Language Models (MLLMs) excel in solving text-based mathematical problems, but they struggle with mathematical diagrams since they are primarily trained on natural scene images. For humans, visual aids generally enhance problem-solving, but MLLMs perform worse as information shifts from textual to visual modality. This decline is mainly due to their shortcomings in aligning images and text. To tackle aforementioned challenges, we propose Math-PUMA, a methodology focused on Progressive Upward Multimodal Alignment. This approach is designed to improve the mathematical reasoning skills of MLLMs through a three-stage training process, with the second stage being the critical alignment stage. We first enhance the language model's mathematical reasoning capabilities with extensive set of textual mathematical problems. We then construct a multimodal dataset with varying degrees of textual and visual information, creating data pairs by presenting each problem in at least two forms. By leveraging the Kullback-Leibler (KL) divergence of next-token prediction distributions to align visual and textual modalities, consistent problem-solving abilities are ensured. Finally, we utilize multimodal instruction tuning for MLLMs with high-quality multimodal data. Experimental results on multiple mathematical reasoning benchmarks demonstrate that the MLLMs trained with Math-PUMA surpass most open-source MLLMs. Our approach effectively narrows the performance gap for problems presented in different modalities. The code and data are available at: https://github.com/wwzhuang01/Math-PUMA.

Pretrained language models have shown superior performance on many natural language processing tasks, yet they still struggle at multi-step formal reasoning tasks like grade school math problems. One key challenge of finetuning them to solve such math reasoning problems is that many existing datasets only contain one reference solution for each problem, despite the fact that there are often alternative solutions resembling different reasoning paths to the final answer. This way, the finetuned models are biased towards the limited reference solutions, which limits their generalization to unseen examples. To mitigate this issue, we propose to let the model perform sampling during training and learn from both self-sampled fully-correct solutions, which yield the correct answer upon execution, and partially-correct solutions, whose intermediate state matches an intermediate state of a known correct solution. We show that our use of self-sampled correct and partially-correct solutions can benefit learning and help guide the sampling process, leading to more efficient exploration of the solution space. Additionally, we explore various training objectives to support learning from multiple solutions per example and find they greatly affect the performance. Experiments on two math reasoning datasets show the effectiveness of our method compared to learning from a single reference solution with MLE, where we improve PASS@100 from 35.5% to 44.5% for GSM8K, and 27.6% to 36.2% PASS@80 for MathQA. Such improvements are also consistent across different model sizes. Our code is available at https://github.com/microsoft/TraceCodegen.

Large-scale pre-trained language models (PLMs) bring new opportunities to challenging problems, especially those that need high-level intelligence, such as the math word problem (MWPs). However, directly applying existing PLMs to MWPs can fail as the generation process lacks sufficient supervision and thus lacks fast adaptivity as humans. We notice that human reasoning has a dual reasoning framework that consists of an immediate reaction system (system 1) and a delicate reasoning system (system 2), where the entire reasoning is determined by their interaction. This inspires us to develop a cooperative reasoning-induced PLM for solving MWPs, called Cooperative Reasoning (CoRe), resulting in a human-like reasoning architecture with system 1 as the generator and system 2 as the verifier. In our approach, the generator is responsible for generating reasoning paths, and the verifiers are used to supervise the evaluation in order to obtain reliable feedback for the generator. We evaluate our CoRe framework on several mathematical reasoning datasets and achieve decent improvement over state-of-the-art methods, up to 9.6% increase over best baselines. Our codes are available at https://github.com/TianHongZXY/CoRe

We present rStar-Math to demonstrate that small language models (SLMs) can rival or even surpass the math reasoning capability of OpenAI o1, without distillation from superior models. rStar-Math achieves this by exercising "deep thinking" through Monte Carlo Tree Search (MCTS), where a math policy SLM performs test-time search guided by an SLM-based process reward model. rStar-Math introduces three innovations to tackle the challenges in training the two SLMs: (1) a novel code-augmented CoT data sythesis method, which performs extensive MCTS rollouts to generate step-by-step verified reasoning trajectories used to train the policy SLM; (2) a novel process reward model training method that avoids na\"ive step-level score annotation, yielding a more effective process preference model (PPM); (3) a self-evolution recipe in which the policy SLM and PPM are built from scratch and iteratively evolved to improve reasoning capabilities. Through 4 rounds of self-evolution with millions of synthesized solutions for 747k math problems, rStar-Math boosts SLMs' math reasoning to state-of-the-art levels. On the MATH benchmark, it improves Qwen2.5-Math-7B from 58.8% to 90.0% and Phi3-mini-3.8B from 41.4% to 86.4%, surpassing o1-preview by +4.5% and +0.9%. On the USA Math Olympiad (AIME), rStar-Math solves an average of 53.3% (8/15) of problems, ranking among the top 20% the brightest high school math students. Code and data will be available at https://github.com/microsoft/rStar.

Large Language Models (LLMs) have recently achieved remarkable progress in mathematical reasoning. To enable such capabilities, many existing works distill strong reasoning models into long chains of thought or design algorithms to construct high-quality math QA data for training. However, these efforts primarily focus on generating correct reasoning paths and answers, while largely overlooking the validity of the questions themselves. In this work, we propose Math Question Verification (MathQ-Verify), a novel five-stage pipeline designed to rigorously filter ill-posed or under-specified math problems. MathQ-Verify first performs format-level validation to remove redundant instructions and ensure that each question is syntactically well-formed. It then formalizes each question, decomposes it into atomic conditions, and verifies them against mathematical definitions. Next, it detects logical contradictions among these conditions, followed by a goal-oriented completeness check to ensure the question provides sufficient information for solving. To evaluate this task, we use existing benchmarks along with an additional dataset we construct, containing 2,147 math questions with diverse error types, each manually double-validated. Experiments show that MathQ-Verify achieves state-of-the-art performance across multiple benchmarks, improving the F1 score by up to 25 percentage points over the direct verification baseline. It further attains approximately 90% precision and 63% recall through a lightweight model voting scheme. MathQ-Verify offers a scalable and accurate solution for curating reliable mathematical datasets, reducing label noise and avoiding unnecessary computation on invalid questions. Our code and data are available at https://github.com/scuuy/MathQ-Verify.

The art of mathematical reasoning stands as a fundamental pillar of intellectual progress and is a central catalyst in cultivating human ingenuity. Researchers have recently published a plethora of works centered around the task of solving Math Word Problems (MWP) - a crucial stride towards general AI. These existing models are susceptible to dependency on shallow heuristics and spurious correlations to derive the solution expressions. In order to ameliorate this issue, in this paper, we propose a framework for MWP solvers based on the generation of linguistic variants of the problem text. The approach involves solving each of the variant problems and electing the predicted expression with the majority of the votes. We use DeBERTa (Decoding-enhanced BERT with disentangled attention) as the encoder to leverage its rich textual representations and enhanced mask decoder to construct the solution expressions. Furthermore, we introduce a challenging dataset, Psmall{ARAMAWPS}, consisting of paraphrased, adversarial, and inverse variants of selectively sampled MWPs from the benchmark Msmall{AWPS} dataset. We extensively experiment on this dataset along with other benchmark datasets using some baseline MWP solver models. We show that training on linguistic variants of problem statements and voting on candidate predictions improve the mathematical reasoning and robustness of the model. We make our code and data publicly available.

Large Language Models (LLMs) exhibit strong potential in mathematical reasoning, yet their effectiveness is often limited by a shortage of high-quality queries. This limitation necessitates scaling up computational responses through self-generated data, yet current methods struggle due to spurious correlated data caused by ineffective exploration across all reasoning stages. To address such challenge, we introduce MARGE: Improving Math Reasoning with Guided Exploration, a novel method to address this issue and enhance mathematical reasoning through hit-guided exploration. MARGE systematically explores intermediate reasoning states derived from self-generated solutions, enabling adequate exploration and improved credit assignment throughout the reasoning process. Through extensive experiments across multiple backbone models and benchmarks, we demonstrate that MARGE significantly improves reasoning capabilities without requiring external annotations or training additional value models. Notably, MARGE improves both single-shot accuracy and exploration diversity, mitigating a common trade-off in alignment methods. These results demonstrate MARGE's effectiveness in enhancing mathematical reasoning capabilities and unlocking the potential of scaling self-generated training data. Our code and models are available at https://github.com/georgao35/MARGE{this link}.

Solving tabular math word problems (TMWPs) has become a critical role in evaluating the mathematical reasoning ability of large language models (LLMs), where large-scale TMWP samples are commonly required for LLM fine-tuning. Since the collection of high-quality TMWP datasets is costly and time-consuming, recent research has concentrated on automatic TMWP generation. However, current generated samples usually suffer from issues of either correctness or diversity. In this paper, we propose a Template-driven LLM-paraphrased (TeLL) framework for generating high-quality TMWP samples with diverse backgrounds and accurate tables, questions, answers, and solutions. To this end, we first extract templates from existing real samples to generate initial problems, ensuring correctness. Then, we adopt an LLM to extend templates and paraphrase problems, obtaining diverse TMWP samples. Furthermore, we find the reasoning annotation is important for solving TMWPs. Therefore, we propose to enrich each solution with illustrative reasoning steps. Through the proposed framework, we construct a high-quality dataset TabMWP-TeLL by adhering to the question types in the TabMWP dataset, and we conduct extensive experiments on a variety of LLMs to demonstrate the effectiveness of TabMWP-TeLL in improving TMWP solving performance. The code and data of this paper are available at: https://github.com/Jason8Kang/TELL.

Large Language Models (LLMs) demonstrate strong performance on mathematical problems when prompted with Chain-of-Thought (CoT), yet it remains unclear whether this success stems from search, rote procedures, or rule-consistent reasoning. To address this, we propose modeling CoT as a certain rule-based stochastic process over directed acyclic graphs (DAGs), where nodes represent intermediate derivation states and edges encode rule applications. Within this framework, we introduce logical closeness, a metric that quantifies how well a model's CoT trajectory (i.e., the LLM's final output) adheres to the DAG structure, providing evaluation beyond classical PASS@k metrics. Building on this, we introduce the DAG-MATH CoT format and construct a benchmark that guides LLMs to generate CoT trajectories in this format, thereby enabling the evaluation of their reasoning ability under our framework. Across standard mathematical reasoning datasets, our analysis uncovers statistically significant differences in reasoning fidelity among representative LLM families-even when PASS@k is comparable-highlighting gaps between final-answer accuracy and rule-consistent derivation. Our framework provides a balance between free-form CoT and formal proofs systems, offering actionable diagnostics for LLMs reasoning evaluation. Our benchmark and code are available at: https://github.com/YuanheZ/DAG-MATH-Formatted-CoT.

Recently, deep learning models have made great progress in MWP solving on answer accuracy. However, they are uninterpretable since they mainly rely on shallow heuristics to achieve high performance without understanding and reasoning the grounded math logic. To address this issue and make a step towards interpretable MWP solving, we first construct a high-quality MWP dataset named InterMWP which consists of 11,495 MWPs and annotates interpretable logical formulas based on algebraic knowledge as the grounded linguistic logic of each solution equation. Different from existing MWP datasets, our InterMWP benchmark asks for a solver to not only output the solution expressions but also predict the corresponding logical formulas. We further propose a novel approach with logical prompt and interpretation generation, called LogicSolver. For each MWP, our LogicSolver first retrieves some highly-correlated algebraic knowledge and then passes them to the backbone model as prompts to improve the semantic representations of MWPs. With these improved semantic representations, our LogicSolver generates corresponding solution expressions and interpretable knowledge formulas in accord with the generated solution expressions, simultaneously. Experimental results show that our LogicSolver has stronger logical formula-based interpretability than baselines while achieving higher answer accuracy with the help of logical prompts, simultaneously. The source code and dataset is available at https://github.com/yangzhch6/InterMWP.

Recognizing handwritten mathematical expressions (HMER) is a challenging task due to the inherent two-dimensional structure, varying symbol scales, and complex spatial relationships among symbols. In this paper, we present a self-supervised learning (SSL) framework for HMER that eliminates the need for expensive labeled data. Our approach begins by pretraining an image encoder using a combination of global and local contrastive loss, enabling the model to learn both holistic and fine-grained representations. A key contribution of this work is a novel self-supervised attention network, which is trained using a progressive spatial masking strategy. This attention mechanism is designed to learn semantically meaningful focus regions, such as operators, exponents, and nested mathematical notation, without requiring any supervision. The progressive masking curriculum encourages the network to become increasingly robust to missing or occluded visual information, ultimately improving structural understanding. Our complete pipeline consists of (1) self-supervised pretraining of the encoder, (2) self-supervised attention learning, and (3) supervised fine-tuning with a transformer decoder to generate LATEX sequences. Extensive experiments on CROHME benchmarks demonstrate that our method outperforms existing SSL and fully supervised baselines, validating the effectiveness of our progressive attention mechanism in enhancing HMER performance. Our codebase can be found here.

Visual mathematical reasoning, as a fundamental visual reasoning ability, has received widespread attention from the Large Multimodal Models (LMMs) community. Existing benchmarks, such as MathVista and MathVerse, focus more on the result-oriented performance but neglect the underlying principles in knowledge acquisition and generalization. Inspired by human-like mathematical reasoning, we introduce WE-MATH, the first benchmark specifically designed to explore the problem-solving principles beyond end-to-end performance. We meticulously collect and categorize 6.5K visual math problems, spanning 67 hierarchical knowledge concepts and five layers of knowledge granularity. We decompose composite problems into sub-problems according to the required knowledge concepts and introduce a novel four-dimensional metric, namely Insufficient Knowledge (IK), Inadequate Generalization (IG), Complete Mastery (CM), and Rote Memorization (RM), to hierarchically assess inherent issues in LMMs' reasoning process. With WE-MATH, we conduct a thorough evaluation of existing LMMs in visual mathematical reasoning and reveal a negative correlation between solving steps and problem-specific performance. We confirm the IK issue of LMMs can be effectively improved via knowledge augmentation strategies. More notably, the primary challenge of GPT-4o has significantly transitioned from IK to IG, establishing it as the first LMM advancing towards the knowledge generalization stage. In contrast, other LMMs exhibit a marked inclination towards Rote Memorization - they correctly solve composite problems involving multiple knowledge concepts yet fail to answer sub-problems. We anticipate that WE-MATH will open new pathways for advancements in visual mathematical reasoning for LMMs. The WE-MATH data and evaluation code are available at https://github.com/We-Math/We-Math.

Despite the rapid progress of multimodal large language models (MLLMs), they have largely overlooked the importance of visual processing. In a simple yet revealing experiment, we interestingly find that language-only models, when provided with image captions, can achieve comparable or even better performance than MLLMs that consume raw visual inputs. This suggests that current MLLMs may generate accurate visual descriptions but fail to effectively integrate them during reasoning. Motivated by this, we propose a simple visual perturbation framework that enhances perceptual robustness without requiring algorithmic modifications or additional training data. Our approach introduces three targeted perturbations: distractor concatenation, dominance-preserving mixup, and random rotation, that can be easily integrated into existing post-training pipelines including SFT, DPO, and GRPO. Through extensive experiments across multiple datasets, we demonstrate consistent improvements in mathematical reasoning performance, with gains comparable to those achieved through algorithmic changes. Additionally, we achieve competitive performance among open-source 7B RL-tuned models by training Qwen2.5-VL-7B with visual perturbation. Through comprehensive ablation studies, we analyze the effectiveness of different perturbation strategies, revealing that each perturbation type contributes uniquely to different aspects of visual reasoning. Our findings highlight the critical role of visual perturbation in multimodal mathematical reasoning: better reasoning begins with better seeing. Our code is available at https://github.com/YutingLi0606/Vision-Matters.

We introduce Confucius3-Math, an open-source large language model with 14B parameters that (1) runs efficiently on a single consumer-grade GPU; (2) achieves SOTA performances on a range of mathematical reasoning tasks, outperforming many models with significantly larger sizes. In particular, as part of our mission to enhancing education and knowledge dissemination with AI, Confucius3-Math is specifically committed to mathematics learning for Chinese K-12 students and educators. Built via post-training with large-scale reinforcement learning (RL), Confucius3-Math aligns with national curriculum and excels at solving main-stream Chinese K-12 mathematical problems with low cost. In this report we share our development recipe, the challenges we encounter and the techniques we develop to overcome them. In particular, we introduce three technical innovations: Targeted Entropy Regularization, Recent Sample Recovery and Policy-Specific Hardness Weighting. These innovations encompass a new entropy regularization, a novel data scheduling policy, and an improved group-relative advantage estimator. Collectively, they significantly stabilize the RL training, improve data efficiency, and boost performance. Our work demonstrates the feasibility of building strong reasoning models in a particular domain at low cost. We open-source our model and code at https://github.com/netease-youdao/Confucius3-Math.

Advances in Large Language Models (LLMs) have sparked interest in their ability to solve Olympiad-level math problems. However, the training and evaluation of these models are constrained by the limited size and quality of available datasets, as creating large-scale data for such advanced problems requires extensive effort from human experts. In addition, current benchmarks are prone to contamination, leading to unreliable evaluations. In this paper, we present an automated pipeline that leverages the rich resources of the Art of Problem Solving (AoPS) forum, which predominantly features Olympiad-level problems and community-driven solutions. Using open-source LLMs, we develop a method to extract question-answer pairs from the forum, resulting in AoPS-Instruct, a dataset of more than 600,000 high-quality QA pairs. Our experiments demonstrate that fine-tuning LLMs on AoPS-Instruct improves their reasoning abilities across various benchmarks. Moreover, we build an automatic pipeline that introduces LiveAoPSBench, an evolving evaluation set with timestamps, derived from the latest forum data, providing a contamination-resistant benchmark for assessing LLM performance. Notably, we observe a significant decline in LLM performance over time, suggesting their success on older examples may stem from pre-training exposure rather than true reasoning ability. Our work presents a scalable approach to creating and maintaining large-scale, high-quality datasets for advanced math reasoning, offering valuable insights into the capabilities and limitations of LLMs in this domain. Our benchmark and code is available at https://github.com/DSL-Lab/aops

Mathematical reasoning lies at the heart of artificial intelligence, underpinning applications in education, program verification, and research-level mathematical discovery. Mathematical competitions, in particular, present two challenging problem types: theorem proving, which requires rigorous proofs of stated conclusions, and answer construction, which involves hypothesizing and formally verifying mathematical objects. Large Language Models (LLMs) effectively generate creative candidate answers but struggle with formal verification, while symbolic provers ensure rigor but cannot efficiently handle creative conjecture generation. We introduce the Enumerate-Conjecture-Prove (ECP) framework, a modular neuro-symbolic method integrating LLM-based enumeration and pattern-driven conjecturing with formal theorem proving. We present ConstructiveBench, a dataset of 3,431 answer-construction problems in various math competitions with verified Lean formalizations. On the ConstructiveBench dataset, ECP improves the accuracy of answer construction from a Chain-of-Thought (CoT) baseline of 14.54% to 45.06% with the gpt-4.1-mini model. Moreover, combined with ECP's constructed answers, the state-of-the-art DeepSeek-Prover-V2-7B model generates correct proofs for 858 of the 3,431 constructive problems in Lean, achieving 25.01% accuracy compared to 9.86% for symbolic-only baselines. Our code and dataset are publicly available at https://github.com/JackSun200312/ECP.

Large language models (LLMs) now achieve near-human performance on standard math word problem benchmarks (e.g., GSM8K), yet their true reasoning ability remains disputed. A key concern is that models often produce confident, yet unfounded, answers to unanswerable problems. We introduce TreeCut, a synthetic dataset that systematically generates infinite unanswerable math word problems and their answerable counterparts, by representing each question as a tree and removing chosen necessary conditions. Experiments show TreeCut effectively induce hallucinations in large language models, including GPT-4o and o3-mini, with rates of 64% and 44% in their respective worst-case scenarios under zero-shot setting. Further analysis highlights that deeper or more complex trees, composite item names, and removing necessary condition near the middle of a path all increase the likelihood of hallucinations, underscoring the persistent challenges LLMs face in identifying unanswerable math problems. The dataset generation code and sample data are available at https://github.com/j-bagel/treecut-math.

Large Language Models (LLMs) have demonstrated exceptional capabilities in various natural language tasks, often achieving performances that surpass those of humans. Despite these advancements, the domain of mathematics presents a distinctive challenge, primarily due to its specialized structure and the precision it demands. In this study, we adopted a two-step approach for investigating the proficiency of LLMs in answering mathematical questions. First, we employ the most effective LLMs, as identified by their performance on math question-answer benchmarks, to generate answers to 78 questions from the Math Stack Exchange (MSE). Second, a case analysis is conducted on the LLM that showed the highest performance, focusing on the quality and accuracy of its answers through manual evaluation. We found that GPT-4 performs best (nDCG of 0.48 and P@10 of 0.37) amongst existing LLMs fine-tuned for answering mathematics questions and outperforms the current best approach on ArqMATH3 Task1, considering P@10. Our Case analysis indicates that while the GPT-4 can generate relevant responses in certain instances, it does not consistently answer all questions accurately. This paper explores the current limitations of LLMs in navigating complex mathematical problem-solving. Through case analysis, we shed light on the gaps in LLM capabilities within mathematics, thereby setting the stage for future research and advancements in AI-driven mathematical reasoning. We make our code and findings publicly available for research: https://github.com/gipplab/LLM-Investig-MathStackExchange

Mathematical understanding and reasoning are crucial tasks for assessing the capabilities of artificial intelligence (AI). However, existing benchmarks either require just a few steps of reasoning, or only contain a small amount of data in one specific topic, making it hard to analyse AI's behaviour with reference to different problems within a specific topic in detail. In this work, we propose Conic10K, a challenging math problem dataset on conic sections in Chinese senior high school education. Our dataset contains various problems with different reasoning depths, while only the knowledge from conic sections is required. Since the dataset only involves a narrow range of knowledge, it is easy to separately analyse the knowledge a model possesses and the reasoning ability it has. For each problem, we provide a high-quality formal representation, the reasoning steps, and the final solution. Experiments show that existing large language models, including GPT-4, exhibit weak performance on complex reasoning. We hope that our findings could inspire more advanced techniques for precise natural language understanding and reasoning. Our dataset and codes are available at https://github.com/whyNLP/Conic10K.

Recent work has shown the immense potential of synthetically generated datasets for training large language models (LLMs), especially for acquiring targeted skills. Current large-scale math instruction tuning datasets such as MetaMathQA (Yu et al., 2024) and MAmmoTH (Yue et al., 2024) are constructed using outputs from closed-source LLMs with commercially restrictive licenses. A key reason limiting the use of open-source LLMs in these data generation pipelines has been the wide gap between the mathematical skills of the best closed-source LLMs, such as GPT-4, and the best open-source LLMs. Building on the recent progress in open-source LLMs, our proposed prompting novelty, and some brute-force scaling, we construct OpenMathInstruct-1, a math instruction tuning dataset with 1.8M problem-solution pairs. The dataset is constructed by synthesizing code-interpreter solutions for GSM8K and MATH, two popular math reasoning benchmarks, using the recently released and permissively licensed Mixtral model. Our best model, OpenMath-CodeLlama-70B, trained on a subset of OpenMathInstruct-1, achieves a score of 84.6% on GSM8K and 50.7% on MATH, which is competitive with the best gpt-distilled models. We release our code, models, and the OpenMathInstruct-1 dataset under a commercially permissive license.

Chain-of-Thought (CoT) plays a crucial role in reasoning for math problem solving. We conduct a comprehensive examination of methods for designing CoT, comparing conventional natural language CoT with various program CoTs, including the self-describing program, the comment-describing program, and the non-describing program. Furthermore, we investigate the impact of programming language on program CoTs, comparing Python and Wolfram Language. Through extensive experiments on GSM8K, MATHQA, and SVAMP, we find that program CoTs often have superior effectiveness in math problem solving. Notably, the best performing combination with 30B parameters beats GPT-3.5-turbo by a significant margin. The results show that self-describing program offers greater diversity and thus can generally achieve higher performance. We also find that Python is a better choice of language than Wolfram for program CoTs. The experimental results provide a valuable guideline for future CoT designs that take into account both programming language and coding style for further advancements. Our datasets and code are publicly available.

Using AI to write formal proofs for mathematical problems is a challenging task that has seen some advancements in recent years. Automated systems such as Lean can verify the correctness of proofs written in formal language, yet writing the proofs in formal language can be challenging for humans and machines. The miniF2F benchmark has 20 IMO problems in its test set, yet formal proofs are available only for 6 of these problems (3 of which are only written by mathematicians). The model with best accuracy can only prove 2 of these 20 IMO problems, from 1950s and 60s, while its training set is a secret. In this work, we write complete, original formal proofs for the remaining IMO problems in Lean along with 3 extra problems from IMO 2022 and 2023. This effort expands the availability of proof currently in the public domain by creating 5,880 lines of Lean proof. The goal of the paper is to pave the way for developing AI models that can automatically write the formal proofs for all the IMO problems in miniF2F and beyond by providing an evaluation benchmark. In this pursuit, we devise a method to decompose the proofs of these problems into their building blocks, constructing a dataset of 1,329 lemmas with more than 40k lines of Lean code. These lemmas are not trivial, yet they are approachable, providing the opportunity to evaluate and diagnose the failures and successes of AI models. We evaluate the ability of the SOTA LLMs on our dataset and analyze their success and failure modes from different perspectives. Our dataset and code is available at: https://github.com/roozbeh-yz/IMO-Steps.

We introduce KnowledgeMath, a novel benchmark designed to evaluate LLMs' capabilities in applying financial knowledge to solve complex math word problems. Compared to prior works, this study features three core advancements. First, KnowledgeMath includes 1,259 problems with a hybrid of textual and tabular content and require college-level knowledge in the finance domain for effective resolution. Second, we provide expert-annotated, detailed solution references in Python program format, ensuring a high-quality benchmark for LLM assessment. Finally, we evaluate a wide spectrum of 14 LLMs with different prompting strategies like Chain-of-Thoughts and Program-of-Thoughts. The current best-performing system (i.e., GPT-4 with Program-of-Thoughts) achieves only 45.4% accuracy, leaving substantial room for improvement. While knowledge-augmented LLMs can improve the performance (e.g., from 23.9% to 32.0% for GPT-3.5), it is still significantly lower the estimated human expert performance of 94%. We believe that KnowledgeMath can facilitate future research on domain-specific knowledge retrieval and augmentation into the math word problem-solving process. We will release the benchmark and code at https://github.com/yale-nlp/KnowledgeMath.

Chain-of-thought prompting~(CoT) and tool augmentation have been validated in recent work as effective practices for improving large language models~(LLMs) to perform step-by-step reasoning on complex math-related tasks. However, most existing math reasoning datasets may be not able to fully evaluate and analyze the ability of LLMs in manipulating tools and performing reasoning, as they may only require very few invocations of tools or miss annotations for evaluating intermediate reasoning steps. To address the issue, we construct CARP, a new Chinese dataset consisting of 4,886 computation-intensive algebra problems with formulated annotations on intermediate steps. In CARP, we test four LLMs with CoT prompting, and find that they are all prone to make mistakes at the early steps of the solution, leading to wrong answers. Based on this finding, we propose a new approach that can deliberate the reasoning steps with tool interfaces, namely DELI. In DELI, we first initialize a step-by-step solution based on retrieved exemplars, then iterate two deliberation procedures that check and refine the intermediate steps of the generated solution, from the perspectives of tool manipulation and natural language reasoning, until obtaining converged solutions or reaching the maximum turn. Experimental results on CARP and six other datasets show that the proposed DELI mostly outperforms competitive baselines, and can further boost the performance of existing CoT methods. Our data and code are available in https://github.com/RUCAIBox/CARP.

The math abilities of large language models can represent their abstract reasoning ability. In this paper, we introduce and open-source our math reasoning LLMs InternLM-Math which is continue pre-trained from InternLM2. We unify chain-of-thought reasoning, reward modeling, formal reasoning, data augmentation, and code interpreter in a unified seq2seq format and supervise our model to be a versatile math reasoner, verifier, prover, and augmenter. These abilities can be used to develop the next math LLMs or self-iteration. InternLM-Math obtains open-sourced state-of-the-art performance under the setting of in-context learning, supervised fine-tuning, and code-assisted reasoning in various informal and formal benchmarks including GSM8K, MATH, Hungary math exam, MathBench-ZH, and MiniF2F. Our pre-trained model achieves 30.3 on the MiniF2F test set without fine-tuning. We further explore how to use LEAN to solve math problems and study its performance under the setting of multi-task learning which shows the possibility of using LEAN as a unified platform for solving and proving in math. Our models, codes, and data are released at https://github.com/InternLM/InternLM-Math.

In math reasoning with large language models (LLMs), fine-tuning data augmentation by query evolution and diverse reasoning paths is empirically verified effective, profoundly narrowing the gap between open-sourced LLMs and cutting-edge proprietary LLMs. In this paper, we conduct an investigation for such data augmentation in math reasoning and are intended to answer: (1) What strategies of data augmentation are more effective; (2) What is the scaling relationship between the amount of augmented data and model performance; and (3) Can data augmentation incentivize generalization to out-of-domain mathematical reasoning tasks? To this end, we create a new dataset, AugGSM8K, by complicating and diversifying the queries from GSM8K and sampling multiple reasoning paths. We obtained a series of LLMs called MuggleMath by fine-tuning on subsets of AugGSM8K. MuggleMath substantially achieves new state-of-the-art on GSM8K (from 54% to 68.4% at the scale of 7B, and from 63.9% to 74.0% at the scale of 13B). A log-linear relationship is presented between MuggleMath's performance and the amount of augmented data. We also find that MuggleMath is weak in out-of-domain math reasoning generalization to MATH. This is attributed to the differences in query distribution between AugGSM8K and MATH which suggest that augmentation on a single benchmark could not help with overall math reasoning performance. Codes and AugGSM8K will be uploaded to https://github.com/OFA-Sys/gsm8k-ScRel.

Reasoning capability is pivotal for Large Language Models (LLMs) to solve complex tasks, yet achieving reliable and scalable reasoning remains challenging. While Chain-of-Thought (CoT) prompting has become a mainstream approach, existing methods often suffer from uncontrolled generation, insufficient quality, and limited diversity in reasoning paths. Recent efforts leverage code to enhance CoT by grounding reasoning in executable steps, but such methods are typically constrained to predefined mathematical problems, hindering scalability and generalizability. In this work, we propose Caco (Code-Assisted Chain-of-ThOught), a novel framework that automates the synthesis of high-quality, verifiable, and diverse instruction-CoT reasoning data through code-driven augmentation. Unlike prior work, Caco first fine-tunes a code-based CoT generator on existing math and programming solutions in a unified code format, then scales the data generation to a large amount of diverse reasoning traces. Crucially, we introduce automated validation via code execution and rule-based filtering to ensure logical correctness and structural diversity, followed by reverse-engineering filtered outputs into natural language instructions and language CoTs to enrich task adaptability. This closed-loop process enables fully automated, scalable synthesis of reasoning data with guaranteed executability. Experiments on our created Caco-1.3M dataset demonstrate that Caco-trained models achieve strong competitive performance on mathematical reasoning benchmarks, outperforming existing strong baselines. Further analysis reveals that Caco's code-anchored verification and instruction diversity contribute to superior generalization across unseen tasks. Our work establishes a paradigm for building self-sustaining, trustworthy reasoning systems without human intervention.

Large language models (LLMs) have made impressive progress in handling simple math problems, yet they still struggle with more challenging and complex mathematical tasks. In this paper, we introduce a series of LLMs that employs the Decomposition of thought with code assistance and self-correction for mathematical reasoning, dubbed as DotaMath. DotaMath models tackle complex mathematical tasks by decomposing them into simpler logical subtasks, leveraging code to solve these subtasks, obtaining fine-grained feedback from the code interpreter, and engaging in self-reflection and correction. By annotating diverse interactive tool-use trajectories and employing query evolution on GSM8K and MATH datasets, we generate an instruction fine-tuning dataset called DotaMathQA with 574K query-response pairs. We train a series of base LLMs using imitation learning on DotaMathQA, resulting in DotaMath models that achieve remarkable performance compared to open-source LLMs across various in-domain and out-of-domain benchmarks. Notably, DotaMath-deepseek-7B showcases an outstanding performance of 64.8% on the competitive MATH dataset and 86.7% on GSM8K. Besides, DotaMath-deepseek-7B maintains strong competitiveness on a series of in-domain and out-of-domain benchmarks (Avg. 80.1%). Looking forward, we anticipate that the DotaMath paradigm will open new pathways for addressing intricate mathematical problems. Our code is publicly available at https://github.com/ChengpengLi1003/DotaMath.

We release Code World Model (CWM), a 32-billion-parameter open-weights LLM, to advance research on code generation with world models. To improve code understanding beyond what can be learned from training on static code alone, we mid-train CWM on a large amount of observation-action trajectories from Python interpreter and agentic Docker environments, and perform extensive multi-task reasoning RL in verifiable coding, math, and multi-turn software engineering environments. With CWM, we provide a strong testbed for researchers to explore the opportunities world modeling affords for improving code generation with reasoning and planning in computational environments. We present first steps of how world models can benefit agentic coding, enable step-by-step simulation of Python code execution, and show early results of how reasoning can benefit from the latter. CWM is a dense, decoder-only LLM trained with a context size of up to 131k tokens. Independent of its world modeling capabilities, CWM offers strong performance on general coding and math tasks: it reaches pass@1 scores of 65.8% on SWE-bench Verified (with test-time scaling), 68.6% on LiveCodeBench, 96.6% on Math-500, and 76.0% on AIME 2024. To support further research on code world modeling, we release model checkpoints after mid-training, SFT, and RL.

While a lot of recent research focuses on enhancing the textual reasoning capabilities of Large Language Models (LLMs) by optimizing the multi-agent framework or reasoning chains, several benchmark tasks can be solved with 100% success through direct coding, which is more scalable and avoids the computational overhead associated with textual iterating and searching. Textual reasoning has inherent limitations in solving tasks with challenges in math, logics, optimization, and searching, which is unlikely to be solved by simply scaling up the model and data size. The recently released OpenAI GPT Code Interpreter and multi-agent frameworks such as AutoGen have demonstrated remarkable proficiency of integrating code generation and execution to solve complex tasks using LLMs. However, based on our experiments on 7 existing popular methods for steering code/text generation in both single- and multi-turn settings with 14 tasks and 6 types of LLMs (including the new O1-preview), currently there is no optimal method to correctly steer LLMs to write code when needed. We discover some interesting patterns on when models use code vs. textual reasoning with the evolution to task complexity and model sizes, which even result in an astonishingly inverse scaling law. We also discover that results from LLM written code are not always better than using textual reasoning, even if the task could be solved through code. To mitigate the above issues, we propose three methods to better steer LLM code/text generation and achieve a notable improvement. The costs of token lengths and runtime are thoroughly discussed for all the methods. We believe the problem of steering LLM code/text generation is critical for future research and has much space for further improvement. Project Page, Datasets, and Codes are available at https://yongchao98.github.io/CodeSteer/.

Pretraining large language models (LLMs) on high-quality, structured data such as mathematics and code substantially enhances reasoning capabilities. However, existing math-focused datasets built from Common Crawl suffer from degraded quality due to brittle extraction heuristics, lossy HTML-to-text conversion, and the failure to reliably preserve mathematical structure. In this work, we introduce Nemotron-CC-Math, a large-scale, high-quality mathematical corpus constructed from Common Crawl using a novel, domain-agnostic pipeline specifically designed for robust scientific text extraction. Unlike previous efforts, our pipeline recovers math across various formats (e.g., MathJax, KaTeX, MathML) by leveraging layout-aware rendering with lynx and a targeted LLM-based cleaning stage. This approach preserves the structural integrity of equations and code blocks while removing boilerplate, standardizing notation into LaTeX representation, and correcting inconsistencies. We collected a large, high-quality math corpus, namely Nemotron-CC-Math-3+ (133B tokens) and Nemotron-CC-Math-4+ (52B tokens). Notably, Nemotron-CC-Math-4+ not only surpasses all prior open math datasets-including MegaMath, FineMath, and OpenWebMath-but also contains 5.5 times more tokens than FineMath-4+, which was previously the highest-quality math pretraining dataset. When used to pretrain a Nemotron-T 8B model, our corpus yields +4.8 to +12.6 gains on MATH and +4.6 to +14.3 gains on MBPP+ over strong baselines, while also improving general-domain performance on MMLU and MMLU-Stem. We present the first pipeline to reliably extract scientific content--including math--from noisy web-scale data, yielding measurable gains in math, code, and general reasoning, and setting a new state of the art among open math pretraining corpora. To support open-source efforts, we release our code and datasets.

Large Language Models (LLMs) have shown strong performance in solving mathematical problems, with code-based solutions proving particularly effective. However, the best practice to leverage coding instruction data to enhance mathematical reasoning remains underexplored. This study investigates three key questions: (1) How do different coding styles of mathematical code-based rationales impact LLMs' learning performance? (2) Can general-domain coding instructions improve performance? (3) How does integrating textual rationales with code-based ones during training enhance mathematical reasoning abilities? Our findings reveal that code-based rationales with concise comments, descriptive naming, and hardcoded solutions are beneficial, while improvements from general-domain coding instructions and textual rationales are relatively minor. Based on these insights, we propose CoinMath, a learning strategy designed to enhance mathematical reasoning by diversifying the coding styles of code-based rationales. CoinMath generates a variety of code-based rationales incorporating concise comments, descriptive naming conventions, and hardcoded solutions. Experimental results demonstrate that CoinMath significantly outperforms its baseline model, MAmmoTH, one of the SOTA math LLMs.

Large reasoning models (LRMs) have recently shown promise in solving complex math problems when optimized with Reinforcement Learning (RL). But conventional approaches rely on outcome-only rewards that provide sparse feedback, resulting in inefficient optimization process. In this work, we investigate the function of process reward models (PRMs) to accelerate the RL training for LRMs. We propose a novel intrinsic signal-driven generative process evaluation mechanism operating at the thought level to address major bottlenecks in RL-based training. Specifically, instead of requiring PRMs to know how to solve problems, our method uses intrinsic signals in solutions to judge stepwise correctness and aggregate contiguous correct/incorrect steps into coherent 'thought' units. This structured, thought-level rewards enable more reliable credit assignment by reducing ambiguity in step segmentation and alleviating reward hacking. We further introduce a capability-adaptive reward mechanism that dynamically balances exploration and exploitation based on the LRM's current proficiency, guiding learning without stifling creative trial-and-error. These innovations are integrated into a new off-policy RL algorithm, TP-GRPO, which extends grouped proximal optimization with process-based rewards and improves training efficiency. Experiments on 1.5B and 7B parameter LRMs demonstrate that our method achieves higher problem-solving accuracy with significantly fewer training samples than outcome-only reward baselines. The results validate that well-structured process rewards can substantially accelerate LRM optimization in math reasoning tasks. Code is available at https://github.com/cs-holder/tp_grpo.

While knowledge evaluation in large language models has predominantly focused on academic subjects like math and physics, these assessments often fail to capture the practical demands of real-world professions. In this paper, we introduce IndoCareer, a dataset comprising 8,834 multiple-choice questions designed to evaluate performance in vocational and professional certification exams across various fields. With a focus on Indonesia, IndoCareer provides rich local contexts, spanning six key sectors: (1) healthcare, (2) insurance and finance, (3) creative and design, (4) tourism and hospitality, (5) education and training, and (6) law. Our comprehensive evaluation of 27 large language models shows that these models struggle particularly in fields with strong local contexts, such as insurance and finance. Additionally, while using the entire dataset, shuffling answer options generally maintains consistent evaluation results across models, but it introduces instability specifically in the insurance and finance sectors.

Large language models have demonstrated impressive capabilities across various natural language processing tasks, especially in solving mathematical problems. However, large language models are not good at math theorem proving using formal languages like Lean. A significant challenge in this area is the scarcity of training data available in these formal languages. To address this issue, we propose a novel pipeline that iteratively generates and filters synthetic data to translate natural language mathematical problems into Lean 4 statements, and vice versa. Our results indicate that the synthetic data pipeline can provide useful training data and improve the performance of LLMs in translating and understanding complex mathematical problems and proofs. Our final dataset contains about 57K formal-informal question pairs along with searched proof from the math contest forum and 21 new IMO questions. We open-source our code at https://github.com/InternLM/InternLM-Math and our data at https://huggingface.co/datasets/InternLM/Lean-Workbook.

We present a neural transducer model with visual attention that learns to generate LaTeX markup of a real-world math formula given its image. Applying sequence modeling and transduction techniques that have been very successful across modalities such as natural language, image, handwriting, speech and audio; we construct an image-to-markup model that learns to produce syntactically and semantically correct LaTeX markup code over 150 words long and achieves a BLEU score of 89%; improving upon the previous state-of-art for the Im2Latex problem. We also demonstrate with heat-map visualization how attention helps in interpreting the model and can pinpoint (detect and localize) symbols on the image accurately despite having been trained without any bounding box data.

Large language models (LLMs) have shown increasing competence in solving mathematical reasoning problems. However, many open-source LLMs still struggle with errors in calculation and semantic understanding during intermediate reasoning steps. In this work, we introduce Prove, a simple yet effective framework that leverages translated programs derived from natural language solutions as a verification mechanism to filter out potentially incorrect reasoning paths before aggregating final answers. Unlike vanilla majority voting, our approach filters out solutions whose corresponding program output is inconsistent with the generated solution, aggregating only those that pass verification. We conducted extensive experiments using 13 open-source LLMs from various model families and sizes, ranging from 0.5B to 13B parameters, across eight mathematical benchmarks. Our results show that Prove consistently outperforms vanilla majority voting as a heuristic for solving mathematical reasoning tasks across all model sizes and datasets, achieving improvements of up to 18% on GSM8K and 8% on MATH-500. Our codes are available at https://github.com/declare-lab/prove.

Large language models (LLMs) have made significant progress in natural language understanding and generation, driven by scalable pretraining and advanced finetuning. However, enhancing reasoning abilities in LLMs, particularly via reinforcement learning from human feedback (RLHF), remains challenging due to the scarcity of high-quality preference data, which is labor-intensive to annotate and crucial for reward model (RM) finetuning. To alleviate this issue, we introduce CodePMP, a scalable preference model pretraining (PMP) pipeline that utilizes a large corpus of synthesized code-preference pairs from publicly available high-quality source code. CodePMP improves RM finetuning efficiency by pretraining preference models on large-scale synthesized code-preference pairs. We evaluate CodePMP on mathematical reasoning tasks (GSM8K, MATH) and logical reasoning tasks (ReClor, LogiQA2.0), consistently showing significant improvements in reasoning performance of LLMs and highlighting the importance of scalable preference model pretraining for efficient reward modeling.

We present DeepSeek-Coder-V2, an open-source Mixture-of-Experts (MoE) code language model that achieves performance comparable to GPT4-Turbo in code-specific tasks. Specifically, DeepSeek-Coder-V2 is further pre-trained from an intermediate checkpoint of DeepSeek-V2 with additional 6 trillion tokens. Through this continued pre-training, DeepSeek-Coder-V2 substantially enhances the coding and mathematical reasoning capabilities of DeepSeek-V2, while maintaining comparable performance in general language tasks. Compared to DeepSeek-Coder-33B, DeepSeek-Coder-V2 demonstrates significant advancements in various aspects of code-related tasks, as well as reasoning and general capabilities. Additionally, DeepSeek-Coder-V2 expands its support for programming languages from 86 to 338, while extending the context length from 16K to 128K. In standard benchmark evaluations, DeepSeek-Coder-V2 achieves superior performance compared to closed-source models such as GPT4-Turbo, Claude 3 Opus, and Gemini 1.5 Pro in coding and math benchmarks.

Mathematical word problem-solving has long been recognized as a complex task for small language models (SLMs). A recent study hypothesized that the smallest model size, needed to achieve over 80% accuracy on the GSM8K benchmark, is 34 billion parameters. To reach this level of performance with smaller models, researcher often train SLMs to generate Python code or use tools to help avoid calculation errors. Additionally, they employ ensembling, where outputs of up to 100 model runs are combined to arrive at a more accurate result. Result selection is done using consensus, majority vote or a separate a verifier model used in conjunction with the SLM. Ensembling provides a substantial boost in accuracy but at a significant cost increase with multiple calls to the model (e.g., Phi-GSM uses top-48 to boost the performance from 68.2 to 81.5). In this work, we present Orca-Math, a 7-billion-parameter SLM based on the Mistral-7B, which achieves 86.81% on GSM8k without the need for multiple model calls or the use of verifiers, code execution or any other external tools. Our approach has the following key elements: (1) A high quality synthetic dataset of 200K math problems created using a multi-agent setup where agents collaborate to create the data, (2) An iterative learning techniques that enables the SLM to practice solving problems, receive feedback on its solutions and learn from preference pairs incorporating the SLM solutions and the feedback. When trained with Supervised Fine-Tuning alone, Orca-Math achieves 81.50% on GSM8k pass@1 metric. With iterative preference learning, Orca-Math achieves 86.81% pass@1. Orca-Math surpasses the performance of significantly larger models such as LLAMA-2-70B, WizardMath-70B, Gemini-Pro, ChatGPT-3.5. It also significantly outperforms other smaller models while using much smaller data (hundreds of thousands vs. millions of problems).

Multimodal Large Language Models (MLLMs) have demonstrated remarkable capabilities in visual mathematical reasoning across various existing benchmarks. However, these benchmarks are predominantly based on clean or processed multimodal inputs, without incorporating the images provided by real-world Kindergarten through 12th grade (K-12) educational users. To address this gap, we introduce MathReal, a meticulously curated dataset comprising 2,000 mathematical questions with images captured by handheld mobile devices in authentic scenarios. Each question is an image, containing the question text and visual element. We systematically classify the real images into three primary categories: image quality degradation, perspective variation, and irrelevant content interference, which are further delineated into 14 subcategories. Additionally, MathReal spans five core knowledge and ability categories, which encompass three question types and are divided into three difficulty levels. To comprehensively evaluate the multimodal mathematical reasoning abilities of state-of-the-art MLLMs in real-world scenarios, we design six experimental settings that enable a systematic analysis of their performance. Through extensive experimentation, we find that the problem-solving abilities of existing MLLMs are significantly challenged in realistic educational contexts. Based on this, we conduct a thorough analysis of their performance and error patterns, providing insights into their recognition, comprehension, and reasoning capabilities, and outlining directions for future improvements. Data and code: https://github.com/junfeng0288/MathReal.

Recent studies have shown that large language models' (LLMs) mathematical problem-solving capabilities can be enhanced by integrating external tools, such as code interpreters, and employing multi-turn Chain-of-Thought (CoT) reasoning. While current methods focus on synthetic data generation and Supervised Fine-Tuning (SFT), this paper studies the complementary direct preference learning approach to further improve model performance. However, existing direct preference learning algorithms are originally designed for the single-turn chat task, and do not fully address the complexities of multi-turn reasoning and external tool integration required for tool-integrated mathematical reasoning tasks. To fill in this gap, we introduce a multi-turn direct preference learning framework, tailored for this context, that leverages feedback from code interpreters and optimizes trajectory-level preferences. This framework includes multi-turn DPO and multi-turn KTO as specific implementations. The effectiveness of our framework is validated through training of various language models using an augmented prompt set from the GSM8K and MATH datasets. Our results demonstrate substantial improvements: a supervised fine-tuned Gemma-1.1-it-7B model's performance increased from 77.5% to 83.9% on GSM8K and from 46.1% to 51.2% on MATH. Similarly, a Gemma-2-it-9B model improved from 84.1% to 86.3% on GSM8K and from 51.0% to 54.5% on MATH.

We introduce Stable Code, the first in our new-generation of code language models series, which serves as a general-purpose base code language model targeting code completion, reasoning, math, and other software engineering-based tasks. Additionally, we introduce an instruction variant named Stable Code Instruct that allows conversing with the model in a natural chat interface for performing question-answering and instruction-based tasks. In this technical report, we detail the data and training procedure leading to both models. Their weights are available via Hugging Face for anyone to download and use at https://huggingface.co/stabilityai/stable-code-3b and https://huggingface.co/stabilityai/stable-code-instruct-3b. This report contains thorough evaluations of the models, including multilingual programming benchmarks, and the MT benchmark focusing on multi-turn dialogues. At the time of its release, Stable Code is the state-of-the-art open model under 3B parameters and even performs comparably to larger models of sizes 7 billion and 15 billion parameters on the popular Multi-PL benchmark. Stable Code Instruct also exhibits state-of-the-art performance on the MT-Bench coding tasks and on Multi-PL completion compared to other instruction tuned models. Given its appealing small size, we also provide throughput measurements on a number of edge devices. In addition, we open source several quantized checkpoints and provide their performance metrics compared to the original model.

The advent of pre-trained code language models (CodeLMs) has lead to significant progress in language-to-code generation. State-of-the-art approaches in this area combine CodeLM decoding with sample pruning and reranking using test cases or heuristics based on the execution results. However, it is challenging to obtain test cases for many real-world language-to-code applications, and heuristics cannot well capture the semantic features of the execution results, such as data type and value range, which often indicates the correctness of the program. In this work, we propose LEVER, a simple approach to improve language-to-code generation by learning to verify the generated programs with their execution results. Specifically, we train verifiers to determine whether a program sampled from the CodeLM is correct or not based on the natural language input, the program itself and its execution results. The sampled programs are reranked by combining the verification score with the CodeLM generation probability, and marginalizing over programs with the same execution results. On four datasets across the domains of table QA, math QA and basic Python programming, LEVER consistently improves over the base CodeLMs (4.6% to 10.9% with code-davinci-002) and achieves new state-of-the-art results on all of them.

Math reasoning is becoming an ever increasing area of focus as we scale large language models. However, even the previously-toughest evals like MATH are now close to saturated by frontier models (90.0% for o1-mini and 86.5% for Gemini 1.5 Pro). We introduce HARP, Human Annotated Reasoning Problems (for Math), consisting of 5,409 problems from the US national math competitions (A(J)HSME, AMC, AIME, USA(J)MO). Of these, 4,780 have answers that are automatically check-able (with libraries such as SymPy). These problems range six difficulty levels, with frontier models performing relatively poorly on the hardest bracket of 197 problems (average accuracy 41.1% for o1-mini, and 9.6% for Gemini 1.5 Pro). Our dataset also features multiple choices (for 4,110 problems) and an average of two human-written, ground-truth solutions per problem, offering new avenues of research that we explore briefly. We report evaluations for many frontier models and share some interesting analyses, such as demonstrating that frontier models across families intrinsically scale their inference-time compute for more difficult problems. Finally, we open source all code used for dataset construction (including scraping) and all code for evaluation (including answer checking) to enable future research at: https://github.com/aadityasingh/HARP.

Scaling high-quality tutoring remains a major challenge in education. Due to growing demand, many platforms employ novice tutors who, unlike experienced educators, struggle to address student mistakes and thus fail to seize prime learning opportunities. Our work explores the potential of large language models (LLMs) to close the novice-expert knowledge gap in remediating math mistakes. We contribute Bridge, a method that uses cognitive task analysis to translate an expert's latent thought process into a decision-making model for remediation. This involves an expert identifying (A) the student's error, (B) a remediation strategy, and (C) their intention before generating a response. We construct a dataset of 700 real tutoring conversations, annotated by experts with their decisions. We evaluate state-of-the-art LLMs on our dataset and find that the expert's decision-making model is critical for LLMs to close the gap: responses from GPT4 with expert decisions (e.g., "simplify the problem") are +76% more preferred than without. Additionally, context-sensitive decisions are critical to closing pedagogical gaps: random decisions decrease GPT4's response quality by -97% than expert decisions. Our work shows the potential of embedding expert thought processes in LLM generations to enhance their capability to bridge novice-expert knowledge gaps. Our dataset and code can be found at: https://github.com/rosewang2008/bridge.

Large language models (LLMs) pretrained on vast source code have achieved prominent progress in code intelligence. However, existing code LLMs have two main limitations in terms of architecture and pretraining tasks. First, they often adopt a specific architecture (encoder-only or decoder-only) or rely on a unified encoder-decoder network for different downstream tasks. The former paradigm is limited by inflexibility in applications while in the latter, the model is treated as a single system for all tasks, leading to suboptimal performance on a subset of tasks. Secondly, they often employ a limited set of pretraining objectives which might not be relevant to some downstream tasks and hence result in substantial performance degrade. To address these limitations, we propose ``CodeT5+'', a family of encoder-decoder LLMs for code in which component modules can be flexibly combined to suit a wide range of downstream code tasks. Such flexibility is enabled by our proposed mixture of pretraining objectives to mitigate the pretrain-finetune discrepancy. These objectives cover span denoising, contrastive learning, text-code matching, and causal LM pretraining tasks, on both unimodal and bimodal multilingual code corpora. Furthermore, we propose to initialize CodeT5+ with frozen off-the-shelf LLMs without training from scratch to efficiently scale up our models, and explore instruction-tuning to align with natural language instructions. We extensively evaluate CodeT5+ on over 20 code-related benchmarks in different settings, including zero-shot, finetuning, and instruction-tuning. We observe state-of-the-art (SoTA) model performance on various code-related tasks, such as code generation and completion, math programming, and text-to-code retrieval tasks. Particularly, our instruction-tuned CodeT5+ 16B achieves new SoTA results on HumanEval code generation task against other open code LLMs.

The evaluation of mathematical reasoning capabilities is essential for advancing Artificial General Intelligence (AGI). While Large Language Models (LLMs) have shown impressive performance in solving mathematical problems, existing benchmarks such as GSM8K and MATH present limitations, including narrow problem definitions with specific numbers and reliance on predetermined rules that hinder accurate assessments of reasoning and adaptability. This paper introduces the UTMath Benchmark, which robustly evaluates the models through extensive unit tests. It consists of 1,053 problems across 9 mathematical domains, with over 68 test cases per problem. We propose an innovative evaluation framework inspired by unit testing in software development, focusing on both accuracy and reliability of results. Furthermore, we introduce the Reasoning-to-Coding of Thoughts (RCoT) approach, which encourages LLMs to perform explicit reasoning before generating code, leading to generating more advanced solution and improved performance. Furthermore, we are releasing not only the UTMath benchmark but also the UTMath-Train training dataset (more than 70k samples), to support the community in further exploring mathematical reasoning.

Natural language image-caption datasets, widely used for training Large Multimodal Models, mainly focus on natural scenarios and overlook the intricate details of mathematical figures that are critical for problem-solving, hindering the advancement of current LMMs in multimodal mathematical reasoning. To this end, we propose leveraging code as supervision for cross-modal alignment, since code inherently encodes all information needed to generate corresponding figures, establishing a precise connection between the two modalities. Specifically, we co-develop our image-to-code model and dataset with model-in-the-loop approach, resulting in an image-to-code model, FigCodifier and ImgCode-8.6M dataset, the largest image-code dataset to date. Furthermore, we utilize FigCodifier to synthesize novel mathematical figures and then construct MM-MathInstruct-3M, a high-quality multimodal math instruction fine-tuning dataset. Finally, we present MathCoder-VL, trained with ImgCode-8.6M for cross-modal alignment and subsequently fine-tuned on MM-MathInstruct-3M for multimodal math problem solving. Our model achieves a new open-source SOTA across all six metrics. Notably, it surpasses GPT-4o and Claude 3.5 Sonnet in the geometry problem-solving subset of MathVista, achieving improvements of 8.9% and 9.2%. The dataset and models will be released at https://github.com/mathllm/MathCoder.

Recently, large language models (LLMs), especially those that are pretrained on code, have demonstrated strong capabilities in generating programs from natural language inputs in a few-shot or even zero-shot manner. Despite promising results, there is a notable lack of a comprehensive evaluation of these models language-to-code generation capabilities. Existing studies often focus on specific tasks, model architectures, or learning paradigms, leading to a fragmented understanding of the overall landscape. In this work, we present L2CEval, a systematic evaluation of the language-to-code generation capabilities of LLMs on 7 tasks across the domain spectrum of semantic parsing, math reasoning and Python programming, analyzing the factors that potentially affect their performance, such as model size, pretraining data, instruction tuning, and different prompting methods. In addition to assessing model performance, we measure confidence calibration for the models and conduct human evaluations of the output programs. This enables us to identify and analyze the typical failure modes across various tasks and models. L2CEval offers a comprehensive understanding of the capabilities and limitations of LLMs in language-to-code generation. We also release the evaluation framework and all model outputs, hoping to lay the groundwork for further future research in this domain.

While closed-source Large Language Models (LLMs) demonstrate strong mathematical problem-solving abilities, open-source models continue to struggle with such tasks. To bridge this gap, we propose a data augmentation approach and introduce PersonaMathQA, a dataset derived from MATH and GSM8K, on which we train the PersonaMath models. Our approach consists of two stages: the first stage is learning from Persona Diversification, and the second stage is learning from Reflection. In the first stage, we regenerate detailed chain-of-thought (CoT) solutions as instructions using a closed-source LLM and introduce a novel persona-driven data augmentation technique to enhance the dataset's quantity and diversity. In the second stage, we incorporate reflection to fully leverage more challenging and valuable questions. Evaluation of our PersonaMath models on MATH and GSM8K reveals that the PersonaMath-7B model (based on LLaMA-2-7B) achieves an accuracy of 24.2% on MATH and 68.7% on GSM8K, surpassing all baseline methods and achieving state-of-the-art performance. Notably, our dataset contains only 70.3K data points-merely 17.8% of MetaMathQA and 27% of MathInstruct-yet our model outperforms these baselines, demonstrating the high quality and diversity of our dataset, which enables more efficient model training. We open-source the PersonaMathQA dataset, PersonaMath models, and our code for public usage.

Program-of-Thought (PoT) replaces natural language-based Chain-of-Thought (CoT) as the most popular method in Large Language Models (LLMs) mathematical reasoning tasks by utilizing external tool calls to circumvent computational errors. However, our evaluation of the GPT-4 and Llama series reveals that using PoT introduces more reasoning errors, such as incorrect formulas or flawed logic, compared to CoT. To address this issue, we propose Human-Think Language (HTL), which leverages a suite of strategies that help integrate PoT and CoT, encompassing: (1) a new generation paradigm that uses full CoT reasoning to control code generation. (2) Focus Attention, that directs model attention to the CoT reasoning during PoT to generate more logical code. (3) reinforcement learning that utilizes the accuracy of both CoT and PoT responses as rewards to prevent repetitive reasoning steps in LLMs when solving difficult math problems. Our method achieves an average improvement of 6.5% on the Llama-Base model and 4.3% on the Mistral-Base model across 8 mathematical calculation datasets. It also shows significant effectiveness on five out-of-domain datasets by controlling the model's information flow, exhibiting strong transferability. Additionally, HTL shows the most significant improvement in non-mathematical natural language inference task, contributing to a unified reasoning task framework

We demonstrate that a neural network pre-trained on text and fine-tuned on code solves mathematics course problems, explains solutions, and generates new questions at a human level. We automatically synthesize programs using few-shot learning and OpenAI's Codex transformer and execute them to solve course problems at 81% automatic accuracy. We curate a new dataset of questions from MIT's largest mathematics courses (Single Variable and Multivariable Calculus, Differential Equations, Introduction to Probability and Statistics, Linear Algebra, and Mathematics for Computer Science) and Columbia University's Computational Linear Algebra. We solve questions from a MATH dataset (on Prealgebra, Algebra, Counting and Probability, Intermediate Algebra, Number Theory, and Precalculus), the latest benchmark of advanced mathematics problems designed to assess mathematical reasoning. We randomly sample questions and generate solutions with multiple modalities, including numbers, equations, and plots. The latest GPT-3 language model pre-trained on text automatically solves only 18.8% of these university questions using zero-shot learning and 30.8% using few-shot learning and the most recent chain of thought prompting. In contrast, program synthesis with few-shot learning using Codex fine-tuned on code generates programs that automatically solve 81% of these questions. Our approach improves the previous state-of-the-art automatic solution accuracy on the benchmark topics from 8.8% to 81.1%. We perform a survey to evaluate the quality and difficulty of generated questions. This work is the first to automatically solve university-level mathematics course questions at a human level and the first work to explain and generate university-level mathematics course questions at scale, a milestone for higher education.

Mathematical reasoning continues to be a critical challenge in large language model (LLM) development with significant interest. However, most of the cutting-edge progress in mathematical reasoning with LLMs has become closed-source due to lack of access to training data. This lack of data access limits researchers from understanding the impact of different choices for synthesizing and utilizing the data. With the goal of creating a high-quality finetuning (SFT) dataset for math reasoning, we conduct careful ablation experiments on data synthesis using the recently released Llama3.1 family of models. Our experiments show that: (a) solution format matters, with excessively verbose solutions proving detrimental to SFT performance, (b) data generated by a strong teacher outperforms on-policy data generated by a weak student model, (c) SFT is robust to low-quality solutions, allowing for imprecise data filtering, and (d) question diversity is crucial for achieving data scaling gains. Based on these insights, we create the OpenMathInstruct-2 dataset, which consists of 14M question-solution pairs (approx 600K unique questions), making it nearly eight times larger than the previous largest open-source math reasoning dataset. Finetuning the Llama-3.1-8B-Base using OpenMathInstruct-2 outperforms Llama3.1-8B-Instruct on MATH by an absolute 15.9\% (51.9\% rightarrow 67.8\%). Finally, to accelerate the open-source efforts, we release the code, the finetuned models, and the OpenMathInstruct-2 dataset under a commercially permissive license.

Despite recent progress made by large language models in code generation, they still struggle with programs that meet complex requirements. Recent work utilizes plan-and-solve decomposition to decrease the complexity and leverage self-tests to refine the generated program. Yet, planning deep-inside requirements in advance can be challenging, and the tests need to be accurate to accomplish self-improvement. To this end, we propose FunCoder, a code generation framework incorporating the divide-and-conquer strategy with functional consensus. Specifically, FunCoder recursively branches off sub-functions as smaller goals during code generation, represented by a tree hierarchy. These sub-functions are then composited to attain more complex objectives. Additionally, we designate functions via a consensus formed by identifying similarities in program behavior, mitigating error propagation. FunCoder outperforms state-of-the-art methods by +9.8% on average in HumanEval, MBPP, xCodeEval and MATH with GPT-3.5 and GPT-4. Moreover, our method demonstrates superiority on smaller models: With FunCoder, StableCode-3b surpasses GPT-3.5 by +18.6% and achieves 97.7% of GPT-4's performance on HumanEval. Further analysis reveals that our proposed dynamic function decomposition is capable of handling complex requirements, and the functional consensus prevails over self-testing in correctness evaluation.

The research builds and evaluates the adversarial potential to introduce copied code or hallucinated AI recommendations for malicious code in popular code repositories. While foundational large language models (LLMs) from OpenAI, Google, and Anthropic guard against both harmful behaviors and toxic strings, previous work on math solutions that embed harmful prompts demonstrate that the guardrails may differ between expert contexts. These loopholes would appear in mixture of expert's models when the context of the question changes and may offer fewer malicious training examples to filter toxic comments or recommended offensive actions. The present work demonstrates that foundational models may refuse to propose destructive actions correctly when prompted overtly but may unfortunately drop their guard when presented with a sudden change of context, like solving a computer programming challenge. We show empirical examples with trojan-hosting repositories like GitHub, NPM, NuGet, and popular content delivery networks (CDN) like jsDelivr which amplify the attack surface. In the LLM's directives to be helpful, example recommendations propose application programming interface (API) endpoints which a determined domain-squatter could acquire and setup attack mobile infrastructure that triggers from the naively copied code. We compare this attack to previous work on context-shifting and contrast the attack surface as a novel version of "living off the land" attacks in the malware literature. In the latter case, foundational language models can hijack otherwise innocent user prompts to recommend actions that violate their owners' safety policies when posed directly without the accompanying coding support request.

Large language models (LLMs) have significantly advanced natural language understanding and demonstrated strong problem-solving abilities. Despite these successes, most LLMs still struggle with solving mathematical problems due to the intricate reasoning required. This paper investigates the mathematical problem-solving capabilities of LLMs using the newly developed "MathOdyssey" dataset. The dataset includes diverse mathematical problems at high school and university levels, created by experts from notable institutions to rigorously test LLMs in advanced problem-solving scenarios and cover a wider range of subject areas. By providing the MathOdyssey dataset as a resource to the AI community, we aim to contribute to the understanding and improvement of AI capabilities in complex mathematical problem-solving. We conduct benchmarking on open-source models, such as Llama-3 and DBRX-Instruct, and closed-source models from the GPT series and Gemini models. Our results indicate that while LLMs perform well on routine and moderately difficult tasks, they face significant challenges with Olympiad-level problems and complex university-level questions. Our analysis shows a narrowing performance gap between open-source and closed-source models, yet substantial challenges remain, particularly with the most demanding problems. This study highlights the ongoing need for research to enhance the mathematical reasoning of LLMs. The dataset, results, and code are publicly available.

Large language models (LLMs) are displaying emergent abilities for math reasoning tasks,and there is a growing attention on enhancing the ability of open-source LLMs through supervised fine-tuning (SFT).In this paper, we aim to explore a general data strategy for supervised data to help optimize and expand math reasoning ability.Firstly, we determine the ability boundary of reasoning paths augmentation by identifying these paths' minimal optimal set.Secondly, we validate that different abilities of the model can be cumulatively enhanced by Mix of Minimal Optimal Sets of corresponding types of data, while our models MMOS achieve SOTA performance on series base models under much lower construction costs.Besides, we point out GSM-HARD is not really hard and today's LLMs no longer lack numerical robustness.Also, we provide an Auto Problem Generator for robustness testing and educational applications.Our code and data are publicly available at https://github.com/cyzhh/MMOS.

Direct Preference Optimization (DPO) using an implicit reward model has proven to be an effective alternative to reinforcement learning from human feedback (RLHF) for fine-tuning preference aligned large language models (LLMs). However, the overall preference annotations of responses do not fully capture the fine-grained quality of model outputs in complex multi-step reasoning tasks, such as mathematical reasoning. To address this limitation, we introduce a novel algorithm called Step-level Value Preference Optimization (SVPO). Our approach employs Monte Carlo Tree Search (MCTS) to automatically annotate step-level preferences for multi-step reasoning. Furthermore, from the perspective of learning-to-rank, we train an explicit value model to replicate the behavior of the implicit reward model, complementing standard preference optimization. This value model enables the LLM to generate higher reward responses with minimal cost during inference. Experimental results demonstrate that our method achieves state-of-the-art performance on both in-domain and out-of-domain mathematical reasoning benchmarks. Our code is available at https://github.com/MARIO-Math-Reasoning/Super_MARIO.

Code generation, symbolic math reasoning, and other tasks require LLMs to produce outputs that are both syntactically and semantically correct. Constrained LLM generation is a promising direction to enforce adherence to formal grammar, but prior works have empirically observed that strict enforcement of formal constraints often diminishes the reasoning capabilities of LLMs. In this work, we first provide a theoretical explanation for why constraining LLM outputs to very restrictive grammars that only allow syntactically valid final answers reduces the reasoning capabilities of the model. Second, we demonstrate that by augmenting the output grammar with carefully designed additional rules, it is always possible to preserve the reasoning capabilities of the LLM while ensuring syntactic and semantic correctness in its outputs. Building on these theoretical insights, we propose a reasoning-augmented constrained decoding algorithm, CRANE, which effectively balances the correctness of constrained generation with the flexibility of unconstrained generation. Experiments on multiple open-source LLMs and benchmarks show that CRANE significantly outperforms both state-of-the-art constrained decoding strategies and standard unconstrained decoding, showing up to 10% points accuracy improvement over baselines on challenging symbolic reasoning benchmarks GSM-symbolic and FOLIO.

Masked diffusion language models (MDLMs) promise fast, non-autoregressive text generation, yet existing samplers, which pick tokens to unmask based on model confidence, ignore interactions when unmasking multiple positions in parallel and effectively reduce to slow, autoregressive behavior. We propose the Dilated Unmasking Scheduler (DUS), an inference-only, planner-model-free method that partitions sequence positions into non-adjacent dilated groups and unmasked them in parallel so as to minimize an upper bound on joint entropy gain at each denoising step. By explicitly trading off the number of network calls against generation quality, DUS recovers most of the performance lost under traditional parallel unmasking strategies. Across math (GSM8K, MATH500), code (HumanEval, MBPP) and general-knowledge benchmarks (BBH, MMLU-Pro), DUS outperforms confidence-based planners, without modifying the underlying denoiser, and reveals the true speed-quality frontier of MDLMs.

Recent progress, such as DeepSeek-R1, has shown that the GRPO algorithm, a Reinforcement Learning (RL) approach, can effectively train Chain-of-Thought (CoT) reasoning in Large Language Models (LLMs) and Vision-Language Models (VLMs). In this paper, we analyze three challenges of GRPO: gradient coupling between thoughts and answers, sparse reward signals caused by limited parallel sampling, and unstable advantage estimation. To mitigate these challenges, we propose GRPO-MA, a simple yet theoretically grounded method that leverages multi-answer generation from each thought process, enabling more robust and efficient optimization. Theoretically, we show that the variance of thought advantage decreases as the number of answers per thought increases. Empirically, our gradient analysis confirms this effect, showing that GRPO-MA reduces gradient spikes compared to GRPO. Experiments on math, code, and diverse multimodal tasks demonstrate that GRPO-MA substantially improves performance and training efficiency. Our ablation studies further reveal that increasing the number of answers per thought consistently enhances model performance.

We propose a framework for robust evaluation of reasoning capabilities of language models, using functional variants of benchmarks. Models that solve a reasoning test should exhibit no difference in performance over the static version of a problem compared to a snapshot of the functional variant. We have rewritten the relevant fragment of the MATH benchmark into its functional variant MATH(), with functionalization of other benchmarks to follow. When evaluating current state-of-the-art models over snapshots of MATH(), we find a reasoning gap -- the percentage difference between the static and functional accuracies. We find reasoning gaps from 58.35% to 80.31% among the state-of-the-art closed and open weights models that perform well on static benchmarks, with the caveat that the gaps are likely to be smaller with more sophisticated prompting strategies. Here we show that models which anecdotally have good reasoning performance over real-world tasks, have quantifiable lower gaps, motivating the open problem of building "gap 0" models. Code for evaluation and new evaluation datasets, three MATH() snapshots, are publicly available at https://github.com/consequentai/fneval/.

This paper introduces a novel benchmark, EGE-Math Solutions Assessment Benchmark, for evaluating Vision-Language Models (VLMs) on their ability to assess hand-written mathematical solutions. Unlike existing benchmarks that focus on problem solving, our approach centres on understanding student solutions, identifying mistakes, and assigning grades according to fixed criteria. We compile 122 scanned solutions from the Russian Unified State Exam (EGE) together with official expert grades, and evaluate seven modern VLMs from Google, OpenAI, Arcee AI, and Alibaba Cloud in three inference modes. The results reveal current limitations in mathematical reasoning and human-rubric alignment, opening new research avenues in AI-assisted assessment. You can find code in https://github.com/Karifannaa/Auto-check-EGE-math

Mathematical reasoning is critical for tasks such as precise distance and area computations, trajectory estimations, and spatial analysis in unmanned aerial vehicle (UAV) based remote sensing, yet current vision-language models (VLMs) have not been adequately tested in this domain. To address this gap, we introduce AVI-Math, the first benchmark to rigorously evaluate multimodal mathematical reasoning in aerial vehicle imagery, moving beyond simple counting tasks to include domain-specific knowledge in areas such as geometry, logic, and algebra. The dataset comprises 3,773 high-quality vehicle-related questions captured from UAV views, covering 6 mathematical subjects and 20 topics. The data, collected at varying altitudes and from multiple UAV angles, reflects real-world UAV scenarios, ensuring the diversity and complexity of the constructed mathematical problems. In this paper, we benchmark 14 prominent VLMs through a comprehensive evaluation and demonstrate that, despite their success on previous multimodal benchmarks, these models struggle with the reasoning tasks in AVI-Math. Our detailed analysis highlights significant limitations in the mathematical reasoning capabilities of current VLMs and suggests avenues for future research. Furthermore, we explore the use of Chain-of-Thought prompting and fine-tuning techniques, which show promise in addressing the reasoning challenges in AVI-Math. Our findings not only expose the limitations of VLMs in mathematical reasoning but also offer valuable insights for advancing UAV-based trustworthy VLMs in real-world applications. The code, and datasets will be released at https://github.com/VisionXLab/avi-math

Metacognitive knowledge refers to humans' intuitive knowledge of their own thinking and reasoning processes. Today's best LLMs clearly possess some reasoning processes. The paper gives evidence that they also have metacognitive knowledge, including ability to name skills and procedures to apply given a task. We explore this primarily in context of math reasoning, developing a prompt-guided interaction procedure to get a powerful LLM to assign sensible skill labels to math questions, followed by having it perform semantic clustering to obtain coarser families of skill labels. These coarse skill labels look interpretable to humans. To validate that these skill labels are meaningful and relevant to the LLM's reasoning processes we perform the following experiments. (a) We ask GPT-4 to assign skill labels to training questions in math datasets GSM8K and MATH. (b) When using an LLM to solve the test questions, we present it with the full list of skill labels and ask it to identify the skill needed. Then it is presented with randomly selected exemplar solved questions associated with that skill label. This improves accuracy on GSM8k and MATH for several strong LLMs, including code-assisted models. The methodology presented is domain-agnostic, even though this article applies it to math problems.

Large language models (LLMs) empowered by chain-of-thought reasoning have achieved impressive accuracy on complex tasks but suffer from excessive inference costs and latency when applied uniformly to all problems. We propose SABER (Switchable and Balanced Training for Efficient LLM Reasoning), a reinforcement learning framework that endows LLMs with user-controllable, token-budgeted reasoning. SABER first profiles each training example's base-model thinking token usage and assigns it to one of the predefined budget tiers. During fine-tuning, the model is guided by system prompts and length-aware rewards to respect its assigned budget. In parallel, we incorporate no-think examples to ensure the model remains reliable even when explicit reasoning is turned off. SABER further supports four discrete inference modes - NoThink, FastThink, CoreThink, and DeepThink, enabling flexible trade-offs between latency and reasoning depth. Extensive evaluations on math reasoning (MATH, GSM8K), code generation (MBPP), and logical reasoning (LiveBench-Reasoning) demonstrate that SABER achieves high accuracy under tight budgets, graceful degradation, and effective cross-scale and cross-domain generalization. In particular, SABER-FastThink cuts reasoning length by 65.4% and yields a 3.6% accuracy gain compared with the base model on the MATH benchmark.

The process of creating training data to teach models is currently driven by humans, who manually analyze model weaknesses and plan how to create data that improves a student model. Recent approaches using LLMs as annotators reduce human effort, but still require humans to interpret feedback from evaluations and control the LLM to produce data the student needs. Automating this labor-intensive process by creating autonomous data generation agents - or teachers - is desirable, but requires environments that can simulate the feedback-driven, iterative, closed loop of data creation. To enable rapid and scalable testing for such agents and their modules, we introduce DataEnvGym, a testbed of teacher environments for data generation agents. DataEnvGym frames data generation as a sequential decision-making task, involving an agent consisting of a data generation policy (which generates a plan for creating training data) and a data generation engine (which transforms the plan into data), inside an environment that provides student feedback. The agent's goal is to improve student performance. Students are iteratively trained and evaluated on generated data, with their feedback (in the form of errors or weak skills) being reported to the agent after each iteration. DataEnvGym includes multiple teacher environment instantiations across 3 levels of structure in the state representation and action space. More structured environments are based on inferred skills and offer more interpretability and curriculum control. We support 3 diverse tasks (math, code, and VQA) and test multiple students and teachers. Example agents in our teaching environments can iteratively improve students across tasks and settings. Moreover, we show that environments teach different skill levels and test variants of key modules, pointing to future work in improving data generation agents, engines, and feedback mechanisms.

In this work, the Localized Filtering-based Attention (LFA) is introduced to incorporate prior knowledge of local dependencies of natural language into Attention. Based on LFA, we develop and release Yuan 2.0, a large language model with parameters ranging from 2.1 billion to 102.6 billion. A data filtering and generation method is presented to build pretraining and fine-tuning dataset in high quality. A distributed training method with non-uniform pipeline parallel, data parallel, and optimizer parallel is proposed, which greatly reduces the bandwidth requirements of intra-node communication, and achieves good performance in large-scale distributed training. Yuan 2.0 models display impressive ability in code generation, math problem-solving, and chat compared with existing models. The latest version of YUAN 2.0, including model weights and source code, is accessible at Github.

Critical thinking is essential for rational decision-making and problem-solving. This skill hinges on the ability to provide precise and reasoned critiques and is a hallmark of human intelligence. In the era of large language models (LLMs), this study explores the ability of LLMs to deliver accurate critiques across various tasks. We are interested in this topic as a capable critic model could not only serve as a reliable evaluator, but also as a source of supervised signals for model tuning. Particularly, if a model can self-critique, it has the potential for autonomous self-improvement. To examine this, we introduce a unified evaluation framework for assessing the critique abilities of LLMs. We develop a benchmark called CriticBench, which comprises 3K high-quality natural language queries and corresponding model responses; and annotate the correctness of these responses. The benchmark cover tasks such as math problem-solving, code completion, and question answering. We evaluate multiple LLMs on the collected dataset and our analysis reveals several noteworthy insights: (1) Critique is generally challenging for most LLMs, and this capability often emerges only when models are sufficiently large. (2) In particular, self-critique is especially difficult. Even top-performing LLMs struggle to achieve satisfactory performance. (3) Models tend to have lower critique accuracy on problems where they are most uncertain. To this end, we introduce a simple yet effective baseline named self-check, which leverages self-critique to improve task performance for various models. We hope this study serves as an initial exploration into understanding the critique abilities of LLMs, and aims to inform future research, including the development of more proficient critic models and the application of critiques across diverse tasks.

Recent advancements in Multi-Agent Systems (MAS) powered by Large Language Models (LLMs) have demonstrated tremendous potential in diverse task scenarios. Nonetheless, existing agentic systems typically rely on predefined agent-role design spaces and static communication structures, limiting their adaptability as well as flexibility in complex interaction environments and leading to subpar performance on highly specialized and expert-level tasks. To address these issues, we introduce HALO, a multi-agent collaboration framework based on a hierarchical reasoning architecture. Specifically, we incorporate a high-level planning agent for task decomposition, mid-level role-design agents for subtask-specific agent instantiation, and low-level inference agents for subtask execution. Particularly, subtask execution is reformulated as a structured workflow search problem, where Monte Carlo Tree Search (MCTS) systematically explores the agentic action space to construct optimal reasoning trajectories. Additionally, as the majority of users lack expertise in prompt engineering, we leverage an Adaptive Prompt Refinement module to transform raw queries into task-specific prompts. Empirical evaluations on Code Generation (HumanEval), General Reasoning (MMLU), and Arithmetic Reasoning (MATH) benchmark datasets highlight the effectiveness of HALO, yielding a 14.4% average improvement over state-of-the-art baselines. Notably, HALO achieves up to 13.3% performance gain on the Moral Scenarios subject in the MMLU benchmark and up to 19.6% performance gain on the Algebra subarea in the MATH benchmark, indicating its advanced proficiency in tackling highly specialized and expert-level tasks. The code repository is available at https://github.com/23japhone/HALO.

Language models (LMs) can solve tasks such as answering questions about tables or images by writing programs. However, using primitive functions often leads to verbose and error-prone programs, and higher-level functions require expert design. To enable better solutions without human labor, we ask code LMs to curate reusable high-level functions, and use them to write solutions. We present TROVE, a training-free method of inducing a verifiable and efficient toolbox of functions, by generating via using, growing, and periodically trimming the toolbox. On 11 datasets from math, table question answering, and image reasoning tasks, TROVE consistently yields simpler solutions with higher accuracy than baselines using CODELLAMA and previous methods using GPT, while using 79-98% smaller toolboxes. TROVE further enables 31% faster and 13% more accurate human verification than baselines. With the same pipeline, it creates diverse functions for varied tasks and datasets, providing insights into their individual characteristics.

Language model heavily depends on high-quality data for optimal performance. Existing approaches rely on manually designed heuristics, the perplexity of existing models, training classifiers, or careful prompt engineering, which require significant expert experience and human annotation effort while introduce biases. We introduce CritiQ, a novel data selection method that automatically mines criteria from human preferences for data quality with only sim30 human-annotated pairs and performs efficient data selection. The main component, CritiQ Flow, employs a manager agent to evolve quality criteria and worker agents to make pairwise judgments. We build a knowledge base that extracts quality criteria from previous work to boost CritiQ Flow. Compared to perplexity- and classifier- based methods, verbal criteria are more interpretable and possess reusable value. After deriving the criteria, we train the CritiQ Scorer to give quality scores and perform efficient data selection. We demonstrate the effectiveness of our method in the code, math, and logic domains, achieving high accuracy on human-annotated test sets. To validate the quality of the selected data, we continually train Llama 3.1 models and observe improved performance on downstream tasks compared to uniform sampling. Ablation studies validate the benefits of the knowledge base and the reflection process. We analyze how criteria evolve and the effectiveness of majority voting.

Accurate mathematical reasoning with Large Language Models (LLMs) is crucial in revolutionizing domains that heavily rely on such reasoning. However, LLMs often encounter difficulties in certain aspects of mathematical reasoning, leading to flawed reasoning and erroneous results. To mitigate these issues, we introduce a novel mechanism, the Chain of Self-Correction (CoSC), specifically designed to embed self-correction as an inherent ability in LLMs, enabling them to validate and rectify their own results. The CoSC mechanism operates through a sequence of self-correction stages. In each stage, the LLMs generate a program to address a given problem, execute this program using program-based tools to obtain an output, subsequently verify this output. Based on the verification, the LLMs either proceed to the next correction stage or finalize the answer. This iterative self-correction process allows the LLMs to refine their reasoning steps and improve the accuracy of their mathematical reasoning. To enable the CoSC mechanism at a low cost, we employ a two-phase finetuning approach. In the first phase, the LLMs are trained with a relatively small volume of seeding data generated from GPT-4, establishing an initial CoSC capability. In the second phase, the CoSC capability is further enhanced by training with a larger volume of self-generated data using the trained model in the first phase, without relying on the paid GPT-4. Our comprehensive experiments demonstrate that CoSC significantly improves performance on traditional mathematical datasets among existing open-source LLMs. Notably, our CoSC-Code-34B model achieved a 53.5% score on MATH, the most challenging mathematical reasoning dataset in the public domain, surpassing the performance of well-established models such as ChatGPT, GPT-4, and even multi-modal LLMs like GPT-4V, Gemini-1.0 Pro, and Gemini-1.0 Ultra.

In the domain of Large Language Model (LLM), LLMs demonstrate significant capabilities in natural language understanding and generation. With the growing needs of applying LLMs on various domains, it is a research question that how to efficiently train and build a model that has expertise in different domains but with a low training cost. We propose CCoE architecture, a framework of easily coupling multiple strong domain experts together to fuse into a big LLM, provides a collective way of utilizing the different domain expert LLMs. Besides, training a large collaborative of multiple expert LLMs requires a high requirements on training sources. CCoE bypasses this problem through isolating other experts and train each expert separately. The design of CCoE assembles multiple expert LLMs through the CoE (Collaboration of Experts) layer. Each CoE layer could have one or more expert LLMs. Expert LLMs have different number of layers and have been well-trained for different domain tasks. Each expert is fine-tuned to be able to achieve the comparable results with SOTA domain LLMs. We start from 5 experts in the domain of Code, Math, Law, text-to-SQL and Medical. The results indicate that our CCoE framework can easily and efficiently boost nearly 10%-20% performance on original base model in different domains but using less resources on training, as well as inference.

We study the task of prompting large-scale language models to perform multi-step reasoning. Existing work shows that when prompted with a chain of thoughts (CoT), sequences of short sentences describing intermediate reasoning steps towards a final answer, large language models can generate new reasoning chains and predict answers for new inputs. A central question is which reasoning examples make the most effective prompts. In this work, we propose complexity-based prompting, a simple and effective example selection scheme for multi-step reasoning. We show that prompts with higher reasoning complexity, i.e., chains with more reasoning steps, achieve substantially better performance on multi-step reasoning tasks over strong baselines. We further extend our complexity-based criteria from prompting (selecting inputs) to decoding (selecting outputs), where we sample multiple reasoning chains from the model, then choose the majority of generated answers from complex reasoning chains (over simple chains). When used to prompt GPT-3 and Codex, our approach substantially improves multi-step reasoning accuracy and achieves new state-of-the-art (SOTA) performance on three math benchmarks (GSM8K, MultiArith, and MathQA) and two BigBenchHard tasks (Date Understanding and Penguins), with an average +5.3 and up to +18 accuracy improvements. Compared with existing example selection schemes like manual tuning or retrieval-based selection, selection based on reasoning complexity is intuitive, easy to implement, and annotation-efficient. Further results demonstrate the robustness of performance gains from complex prompts under format perturbation and distribution shift.

A key challenge in applying reinforcement learning (RL) to diffusion large language models (dLLMs) lies in the intractability of their likelihood functions, which are essential for the RL objective, necessitating corresponding approximation in each training step. While existing methods approximate the log-likelihoods by their evidence lower bounds (ELBOs) via customized Monte Carlo (MC) sampling, the forward computational graphs of all MC samples need to be retained for the gradient computation of non-linear terms in the RL objective, resulting in significant memory overhead. This constraint restricts feasible sample sizes, leading to imprecise likelihood approximations and ultimately distorting the RL objective. To overcome this limitation, we propose Boundary-Guided Policy Optimization (BGPO), a memory-efficient RL algorithm that maximizes a specially constructed lower bound of the ELBO-based objective. This lower bound is carefully designed to satisfy two key properties: (1) Linearity: it is formulated in a linear sum where each term depends only on a single MC sample, thereby enabling gradient accumulation across samples and ensuring constant memory usage; (2) Equivalence: Both the value and gradient of this lower bound are equal to those of the ELBO-based objective in on-policy training, making it also an effective approximation for the original RL objective. These properties allow BGPO to adopt a large MC sample size, resulting in more accurate likelihood approximations and improved RL objective estimation, which in turn leads to enhanced performance. Experiments show that BGPO significantly outperforms previous RL algorithms for dLLMs in math problem solving, code generation, and planning tasks.

While large language models (LLMs) have achieved remarkable reasoning capabilities across domains like code, math and other enterprise tasks, their significant memory and computational costs often preclude their use in practical enterprise settings. To this end, we introduce Apriel-Nemotron-15B-Thinker, a 15-billion parameter model in the ServiceNow Apriel SLM series that achieves performance against medium sized state-of-the-art models such as o1-mini, QWQ32B, and EXAONE-Deep-32B while maintaining only half the memory footprint of those alternatives. Apriel-Nemotron-15B-Thinker model is trained in a four stage training pipeline including 1) Base Model upscaling, 2) Continual Pre-training 3) Supervised Fine-tuning (SFT) and 4) Reinforcement Learning using GRPO. Comprehensive evaluations across a diverse suite of benchmarks consistently demonstrate that our Apriel-Nemotron-15B-Thinker model matches or exceeds the performance of its 32-billion parameter counterparts, despite being less than half their size.

Enabling large language models (LLMs) to perform tasks in zero-shot has been an appealing goal owing to its labor-saving (i.e., requiring no task-specific annotations); as such, zero-shot prompting approaches also enjoy better task generalizability. To improve LLMs' zero-shot performance, prior work has focused on devising more effective task instructions (e.g., ``let's think step by step'' ). However, we argue that, in order for an LLM to solve them correctly in zero-shot, individual test instances need more carefully designed and customized instructions. To this end, we propose PRoMPTd, an approach that rewrites the task prompt for each individual test input to be more specific, unambiguous, and complete, so as to provide better guidance to the task LLM. We evaluated PRoMPTd on eight datasets covering tasks including arithmetics, logical reasoning, and code generation, using GPT-4 as the task LLM. Notably, PRoMPTd achieves an absolute improvement of around 10% on the complex MATH dataset and 5% on the code generation task on HumanEval, outperforming conventional zero-shot methods. In addition, we also showed that the rewritten prompt can provide better interpretability of how the LLM resolves each test instance, which can potentially be leveraged as a defense mechanism against adversarial prompting. The source code and dataset can be obtained from https://github.com/salokr/PRoMPTd

We show that reinforcement learning with verifiable rewards (RLVR) can elicit strong mathematical reasoning in certain models even with spurious rewards that have little, no, or even negative correlation with the correct answer. For example, RLVR improves MATH-500 performance for Qwen2.5-Math-7B in absolute points by 21.4% (random reward), 13.8% (format reward), 24.1% (incorrect label), 26.0% (1-shot RL), and 27.1% (majority voting) -- nearly matching the 29.1% gained with ground truth rewards. However, the spurious rewards that work for Qwen often fail to yield gains with other model families like Llama3 or OLMo2. In particular, we find code reasoning -- thinking in code without actual code execution -- to be a distinctive Qwen2.5-Math behavior that becomes significantly more frequent after RLVR, from 65% to over 90%, even with spurious rewards. Overall, we hypothesize that, given the lack of useful reward signal, RLVR must somehow be surfacing useful reasoning representations learned during pretraining, although the exact mechanism remains a topic for future work. We suggest that future RLVR research should possibly be validated on diverse models rather than a single de facto choice, as we show that it is easy to get significant performance gains on Qwen models even with completely spurious reward signals.

In the context of multi-step reasoning, language models (LMs) probabilities are often miscalibrated -- solutions with high probabilities are not always correct. Therefore, greedy decoding, which is the standard decoding method for reasoning tasks, often yields incorrect solutions. In addition, methods such as self-consistency and verifiers rely on sampling from the LM distribution and do not tackle the underlying issue. To address this, we introduce Guiding Multi-step ReAsoning with a CorrectnEss Discriminator (GRACE), a stepwise decoding approach that nudges the model towards producing correct reasoning steps. GRACE employs a discriminator model, which is trained to differentiate correct steps from invalid ones, to adjust decoding preferences based on the correctness of each reasoning step. Importantly, GRACE does not require fine-tuning or re-training the LMs. When compared with conventional decoding strategies over four popular math reasoning benchmarks, GRACE exhibits significant improvements in both final answer accuracy and step correctness, outperforming both greedy decoding and self-consistency.Our code can be found at \url{https://github.com/mukhal/grace.}

Large language models (LLMs) have recently demonstrated an impressive ability to perform arithmetic and symbolic reasoning tasks, when provided with a few examples at test time ("few-shot prompting"). Much of this success can be attributed to prompting methods such as "chain-of-thought'', which employ LLMs for both understanding the problem description by decomposing it into steps, as well as solving each step of the problem. While LLMs seem to be adept at this sort of step-by-step decomposition, LLMs often make logical and arithmetic mistakes in the solution part, even when the problem is decomposed correctly. In this paper, we present Program-Aided Language models (PAL): a novel approach that uses the LLM to read natural language problems and generate programs as the intermediate reasoning steps, but offloads the solution step to a runtime such as a Python interpreter. With PAL, decomposing the natural language problem into runnable steps remains the only learning task for the LLM, while solving is delegated to the interpreter. We demonstrate this synergy between a neural LLM and a symbolic interpreter across 13 mathematical, symbolic, and algorithmic reasoning tasks from BIG-Bench Hard and other benchmarks. In all these natural language reasoning tasks, generating code using an LLM and reasoning using a Python interpreter leads to more accurate results than much larger models. For example, PAL using Codex achieves state-of-the-art few-shot accuracy on the GSM8K benchmark of math word problems, surpassing PaLM-540B which uses chain-of-thought by absolute 15% top-1. Our code and data are publicly available at http://reasonwithpal.com/ .

Humans are good at learning on the job: We learn how to solve the tasks we face as we go along. Can a model do the same? We propose an agent that assembles a task-specific curriculum, called test-time curriculum (TTC-RL), and applies reinforcement learning to continue training the model for its target task. The test-time curriculum avoids time-consuming human curation of datasets by automatically selecting the most task-relevant data from a large pool of available training data. Our experiments demonstrate that reinforcement learning on a test-time curriculum consistently improves the model on its target tasks, across a variety of evaluations and models. Notably, on challenging math and coding benchmarks, TTC-RL improves the pass@1 of Qwen3-8B by approximately 1.8x on AIME25 and 2.1x on CodeElo. Moreover, we find that TTC-RL significantly raises the performance ceiling compared to the initial model, increasing pass@8 on AIME25 from 40% to 62% and on CodeElo from 28% to 43%. Our findings show the potential of test-time curricula in extending the test-time scaling paradigm to continual training on thousands of task-relevant experiences during test-time.

Large reasoning models (LRMs) like OpenAI-o1 and DeepSeek-R1 have demonstrated remarkable capabilities in complex reasoning tasks through the utilization of long Chain-of-thought (CoT). However, these models often suffer from hallucinations and inefficiencies due to their reliance solely on internal reasoning processes. In this paper, we introduce START (Self-Taught Reasoner with Tools), a novel tool-integrated long CoT reasoning LLM that significantly enhances reasoning capabilities by leveraging external tools. Through code execution, START is capable of performing complex computations, self-checking, exploring diverse methods, and self-debugging, thereby addressing the limitations of LRMs. The core innovation of START lies in its self-learning framework, which comprises two key techniques: 1) Hint-infer: We demonstrate that inserting artificially designed hints (e.g., ``Wait, maybe using Python here is a good idea.'') during the inference process of a LRM effectively stimulates its ability to utilize external tools without the need for any demonstration data. Hint-infer can also serve as a simple and effective sequential test-time scaling method; 2) Hint Rejection Sampling Fine-Tuning (Hint-RFT): Hint-RFT combines Hint-infer and RFT by scoring, filtering, and modifying the reasoning trajectories with tool invocation generated by a LRM via Hint-infer, followed by fine-tuning the LRM. Through this framework, we have fine-tuned the QwQ-32B model to achieve START. On PhD-level science QA (GPQA), competition-level math benchmarks (AMC23, AIME24, AIME25), and the competition-level code benchmark (LiveCodeBench), START achieves accuracy rates of 63.6%, 95.0%, 66.7%, 47.1%, and 47.3%, respectively. It significantly outperforms the base QwQ-32B and achieves performance comparable to the state-of-the-art open-weight model R1-Distill-Qwen-32B and the proprietary model o1-Preview.

Large language models (LLMs) have revolutionized the field of artificial intelligence, enabling natural language processing tasks that were previously thought to be exclusive to humans. In this work, we introduce Qwen, the first installment of our large language model series. Qwen is a comprehensive language model series that encompasses distinct models with varying parameter counts. It includes Qwen, the base pretrained language models, and Qwen-Chat, the chat models finetuned with human alignment techniques. The base language models consistently demonstrate superior performance across a multitude of downstream tasks, and the chat models, particularly those trained using Reinforcement Learning from Human Feedback (RLHF), are highly competitive. The chat models possess advanced tool-use and planning capabilities for creating agent applications, showcasing impressive performance even when compared to bigger models on complex tasks like utilizing a code interpreter. Furthermore, we have developed coding-specialized models, Code-Qwen and Code-Qwen-Chat, as well as mathematics-focused models, Math-Qwen-Chat, which are built upon base language models. These models demonstrate significantly improved performance in comparison with open-source models, and slightly fall behind the proprietary models.

Despite the general capabilities of large pretrained language models, they consistently benefit from further adaptation to better achieve desired behaviors. However, tuning these models has become increasingly resource-intensive, or impossible when model weights are private. We introduce proxy-tuning, a lightweight decoding-time algorithm that operates on top of black-box LMs to achieve the result of directly tuning the model, but by accessing only its prediction over the output vocabulary. Our method instead tunes a smaller LM, then applies the difference between the predictions of the small tuned and untuned LMs to shift the original predictions of the base model in the direction of tuning, while retaining the benefits of larger scale pretraining. In experiments, when we apply proxy-tuning to Llama2-70B using proxies of only 7B size, we can close 88% of the gap between Llama2-70B and its truly-tuned chat version, when evaluated across knowledge, reasoning, and safety benchmarks. Interestingly, when tested on TruthfulQA, proxy-tuned models are actually more truthful than directly tuned models, possibly because decoding-time guidance better retains the model's factual knowledge. We then demonstrate the generality of proxy-tuning by applying it for domain adaptation on code, and task-specific finetuning on question-answering and math problems. Our work demonstrates the promise of using small tuned LMs to efficiently customize large, potentially proprietary LMs through decoding-time guidance.

Data plays the most prominent role in how language models acquire skills and knowledge. The lack of massive, well-organized pre-training datasets results in costly and inaccessible data pipelines. We present Essential-Web v1.0, a 24-trillion-token dataset in which every document is annotated with a twelve-category taxonomy covering topic, format, content complexity, and quality. Taxonomy labels are produced by EAI-Distill-0.5b, a fine-tuned 0.5b-parameter model that achieves an annotator agreement within 3% of Qwen2.5-32B-Instruct. With nothing more than SQL-style filters, we obtain competitive web-curated datasets in math (-8.0% relative to SOTA), web code (+14.3%), STEM (+24.5%) and medical (+8.6%). Essential-Web v1.0 is available on HuggingFace: https://huggingface.co/datasets/EssentialAI/essential-web-v1.0

Grokking, i.e., test performance keeps improving long after training loss converged, has been recently witnessed in neural network training, making the mechanism of generalization and other emerging capabilities such as reasoning mysterious. While prior studies usually train small models on a few toy or highly-specific tasks for thousands of epochs, we conduct the first study of grokking on checkpoints during one-pass pretraining of a 7B large language model (LLM), i.e., OLMoE. We compute the training loss and evaluate generalization on diverse benchmark tasks, including math reasoning, code generation, and commonsense/domain-specific knowledge retrieval tasks. Our study, for the first time, verifies that grokking still happens in the pretraining of large-scale foundation models, though different data may enter grokking stages asynchronously. We further demystify grokking's "emergence of generalization" by investigating LLM internal dynamics. Specifically, we find that training samples' pathways (i.e., expert choices across layers) evolve from random, instance-specific to more structured and shareable between samples during grokking. Also, the complexity of a sample's pathway reduces despite the converged loss. These indicate a memorization-to-generalization conversion, providing a mechanistic explanation of delayed generalization. In the study, we develop two novel metrics to quantify pathway distance and the complexity of a single pathway. We show their ability to predict the generalization improvement on diverse downstream tasks. They are efficient, simple to compute and solely dependent on training data. Hence, they have practical value for pretraining, enabling us to monitor the generalization performance without finetuning and test. Theoretically, we show that more structured pathways reduce model complexity and improve the generalization bound.

Reasoning models have demonstrated impressive performance on difficult tasks that traditional language models struggle at. However, many are plagued with the problem of overthinking--generating large amounts of unnecessary tokens which don't improve accuracy on a question. We introduce approximate measures of problem-level difficulty and demonstrate that a clear relationship between problem difficulty and optimal token spend exists, and evaluate how well calibrated a variety of reasoning models are in terms of efficiently allocating the optimal token count. We find that in general, reasoning models are poorly calibrated, particularly on easy problems. To evaluate calibration on easy questions we introduce DUMB500, a dataset of extremely easy math, reasoning, code, and task problems, and jointly evaluate reasoning model on these simple examples and extremely difficult examples from existing frontier benchmarks on the same task domain. Finally, we introduce THOUGHTTERMINATOR, a training-free black box decoding technique that significantly improves reasoning model calibration.

Open-source Large Language Models (LLMs) increasingly specialize by domain (e.g., math, code, general reasoning), motivating systems that leverage complementary strengths across models. Prior multi-LLM approaches either (i) route a query to one or a few experts and generate independently, (ii) aggregate outputs from each model via costly multi-turn exchanges, or (iii) fuse weights into a single model-typically requiring architectural homogeneity. We introduce Mixture of Thoughts (MoT), a simple method for latent-level collaboration among heterogeneous experts under a global routing scheme. For each query, a lightweight router selects top-K experts and designates a primary expert; uniformly placed interaction layers project hidden states into a shared latent space where the primary expert performs cross-attention over its active (selected) peers. Pre-trained experts remain frozen; only the router and the lightweight interaction layers are trained with a novel joint training objective that improves both the expert selection and inter-expert collaboration. Across five in-distribution (ID) and three out-of-distribution (OOD) benchmarks, MoT surpasses the current routing and aggregation-based state-of-the-art, Avengers, by +0.38% and +2.92%, respectively. Further, MoT significantly outperforms the best-performing single model. It achieves this with single-pass inference, runtime comparable to routing baselines, and none of the overheads of iterative aggregation. MoT offers a simple latent-space mechanism for combining heterogeneous LLMs, a practical step toward broader multi-LLM collaboration. Our code is publicly available at https://github.com/jacobfa/mot.

With the proliferation of domain-specific models, model merging has emerged as a set of techniques that combine the capabilities of multiple models into one that can multitask without the cost of additional training. In this paper, we propose a new model merging technique, Drop and rEscaLe via sampLing with mAgnitude (DELLA-Merging), that employs a novel pruning technique, MAGPRUNE, which shows significant advantages over DARE and TIES. MAGPRUNE first ranks the parameters in order of their magnitude and assigns higher dropout probabilities (p) to parameters with lower ranks corresponding to lower magnitudes. To approximate the original embeddings, MAGPRUNE employs a rescaling operation on the parameters that survive the random dropping by 1/(1 - p). On three different expert models considered for merging (LM, Math, Code) and corresponding benchmark datasets (AlpacaEval, GSM8K, MBPP), DELLA shows an average improvement of 2.4 points over baseline methods employing delta parameter pruning (an improvement of 3.6 points over TIES, 1.2 points over DARE), and 11.1 points over the no-pruning baseline (TA). We release the source code at: https://github.com/declare-lab/della.

Large language models (LLMs) with enormous pre-training tokens and parameter amounts emerge abilities, including math reasoning, code generation, and instruction following. These abilities are further enhanced by supervised fine-tuning (SFT). The open-source community has studied on ad-hoc SFT for each ability, while proprietary LLMs are versatile for all abilities. It is important to investigate how to unlock them with multiple abilities via SFT. In this study, we specifically focus on the data composition between mathematical reasoning, code generation, and general human-aligning abilities during SFT. From a scaling perspective, we investigate the relationship between model abilities and various factors including data amounts, data composition ratio, model parameters, and SFT strategies. Our experiments reveal that different abilities exhibit different scaling patterns, and larger models generally show superior performance with the same amount of data. Mathematical reasoning and code generation improve as data amounts increase consistently, while the general ability is enhanced with about a thousand samples and improves slowly. We find data composition results in various abilities improvements with low data amounts, while conflicts of abilities with high data amounts. Our experiments further show that composition data amount impacts performance, while the influence of composition ratio is insignificant. Regarding the SFT strategies, we evaluate sequential learning multiple abilities are prone to catastrophic forgetting. Our proposed Dual-stage Mixed Fine-tuning (DMT) strategy learns specialized abilities first and then learns general abilities with a small amount of specialized data to prevent forgetting, offering a promising solution to learn multiple abilities with different scaling patterns.

Large pre-trained Transformer models achieve state-of-the-art results across diverse language and reasoning tasks, but full fine-tuning incurs substantial storage, memory, and computational overhead. Parameter-efficient fine-tuning (PEFT) methods mitigate these costs by learning only a small subset of task-specific parameters, yet existing approaches either introduce inference-time latency (adapter modules), suffer from suboptimal convergence (randomly initialized low-rank updates), or rely on fixed rank choices that may not match task complexity (Kronecker-based decompositions). We propose SoKA (SVD on Kronecker Adaptation), a novel PEFT strategy that combines Kronecker-product tensor factorization with SVD-driven initialization and spectrum-aware dynamic rank selection. Our Kronecker-Product SVD (KPSVD) procedure extracts principal components of the full weight update into compact Kronecker factors, while an adaptive rank selection algorithm uses energy-threshold and elbow-point criteria to prune negligible components. Empirical evaluation on LLaMA2-7B across arithmetic reasoning (GSM8K), formal mathematics (MATH), and code generation (MBPP) demonstrates that SoKA requires only 0.99M trainable parameters, 25% fewer than LoRA/PiSSA, while matching or exceeding baseline performance. Moreover, SoKA exhibits faster convergence and more stable gradients, highlighting its robustness and efficiency for large-scale model adaptation.

Fine-tuning is a crucial process for adapting large language models (LLMs) to diverse applications. In certain scenarios, such as multi-tenant serving, deploying multiple LLMs becomes necessary to meet complex demands. Recent studies suggest decomposing a fine-tuned LLM into a base model and corresponding delta weights, which are then compressed using low-rank or low-bit approaches to reduce costs. In this work, we observe that existing low-rank and low-bit compression methods can significantly harm the model performance for task-specific fine-tuned LLMs (e.g., WizardMath for math problems). Motivated by the long-tail distribution of singular values in the delta weights, we propose a delta quantization approach using mixed-precision. This method employs higher-bit representation for singular vectors corresponding to larger singular values. We evaluate our approach on various fine-tuned LLMs, including math LLMs, code LLMs, chat LLMs, and even VLMs. Experimental results demonstrate that our approach performs comparably to full fine-tuned LLMs, surpassing both low-rank and low-bit baselines by a considerable margin. Additionally, we show that our method is compatible with various backbone LLMs, such as Llama-2, Llama-3, and Mistral, highlighting its generalizability.

Multi-agent systems (MAS) have emerged as a powerful paradigm for orchestrating large language models (LLMs) and specialized tools to collaboratively address complex tasks. However, existing MAS frameworks often require manual workflow configuration and lack native support for dynamic evolution and performance optimization. In addition, many MAS optimization algorithms are not integrated into a unified framework. In this paper, we present EvoAgentX, an open-source platform that automates the generation, execution, and evolutionary optimization of multi-agent workflows. EvoAgentX employs a modular architecture consisting of five core layers: the basic components, agent, workflow, evolving, and evaluation layers. Specifically, within the evolving layer, EvoAgentX integrates three MAS optimization algorithms, TextGrad, AFlow, and MIPRO, to iteratively refine agent prompts, tool configurations, and workflow topologies. We evaluate EvoAgentX on HotPotQA, MBPP, and MATH for multi-hop reasoning, code generation, and mathematical problem solving, respectively, and further assess it on real-world tasks using GAIA. Experimental results show that EvoAgentX consistently achieves significant performance improvements, including a 7.44% increase in HotPotQA F1, a 10.00% improvement in MBPP pass@1, a 10.00% gain in MATH solve accuracy, and an overall accuracy improvement of up to 20.00% on GAIA. The source code is available at: https://github.com/EvoAgentX/EvoAgentX

Agentic AI systems, which build on Large Language Models (LLMs) and interact with tools and memory, have rapidly advanced in capability and scope. Yet, since LLMs have been shown to struggle in multilingual settings, typically resulting in lower performance and reduced safety, agentic systems risk inheriting these limitations. This raises concerns about the global accessibility of such systems, as users interacting in languages other than English may encounter unreliable or security-critical agent behavior. Despite growing interest in evaluating agentic AI, existing benchmarks focus exclusively on English, leaving multilingual settings unexplored. To address this gap, we propose MAPS, a multilingual benchmark suite designed to evaluate agentic AI systems across diverse languages and tasks. MAPS builds on four widely used agentic benchmarks - GAIA (real-world tasks), SWE-bench (code generation), MATH (mathematical reasoning), and the Agent Security Benchmark (security). We translate each dataset into ten diverse languages, resulting in 805 unique tasks and 8,855 total language-specific instances. Our benchmark suite enables a systematic analysis of how multilingual contexts affect agent performance and robustness. Empirically, we observe consistent degradation in both performance and security when transitioning from English to other languages, with severity varying by task and correlating with the amount of translated input. Building on these findings, we provide actionable recommendations to guide agentic AI systems development and assessment under multilingual settings. This work establishes a standardized evaluation framework, encouraging future research towards equitable, reliable, and globally accessible agentic AI. MAPS benchmark suite is publicly available at https://huggingface.co/datasets/Fujitsu-FRE/MAPS

The rapid advancement of large language models (LLMs) demands robust, unbiased, and scalable evaluation methods. However, human annotations are costly to scale, model-based evaluations are susceptible to stylistic biases, and target-answer-based benchmarks are vulnerable to data contamination and cheating. To address these limitations, we propose StructTest, a novel benchmark that evaluates LLMs on their ability to follow compositional instructions and generate structured outputs, providing an unbiased, cost-effective, and difficult-to-cheat evaluation framework. Assessments are conducted deterministically using a rule-based evaluator, which can be easily extended to new tasks and datasets. By testing structured outputs across diverse domains including Summarization, Code, HTML, and Math, and evaluating 17 popular LLMs, we demonstrate that StructTest remains challenging even for top-performing models like Deepseek-V3/R1 and GPT-4o, establishing it as a robust proxy for measuring reasoning capabilities. We believe StructTest offers a critical and complementary approach to achieving objective and comprehensive model evaluation.

As current training data for Large Language Models (LLMs) are dominated by English corpus, they are English-centric and they present impressive performance on English reasoning tasks.This paper primarily studies English-centric models, but our method could be universal by using the centric language in the dictionary for non-English-centric LLMs. Yet, they usually suffer from lower performance in other languages. There are about 7,000 languages over the world, and many are low-resourced on English-centric LLMs. For the sake of people who primarily speak these languages, it is especially urgent to enable our LLMs in those languages. Model training is usually effective, but computationally expensive and requires experienced NLP practitioners. This paper presents a novel and simple yet effective method called Dictionary Insertion Prompting (DIP). When providing a non-English prompt, DIP looks up a word dictionary and inserts words' English counterparts into the prompt for LLMs. It then enables better translation into English and better English model thinking steps which leads to obviously better results. We experiment with about 200 languages from FLORES-200. Since there are no adequate datasets, we use the NLLB translator to create synthetic multilingual benchmarks from the existing 4 English reasoning benchmarks such as GSM8K and AQuA. Despite the simplicity and computationally lightweight, we surprisingly found the effectiveness of DIP on math and commonsense reasoning tasks on multiple open-source and close-source LLMs.Our dictionaries, code, and synthetic benchmarks will be open-sourced to facilitate future research.

With the continuous advancement of large language models (LLMs), it is essential to create new benchmarks to effectively evaluate their expanding capabilities and identify areas for improvement. This work focuses on multi-image reasoning, an emerging capability in state-of-the-art LLMs. We introduce ReMI, a dataset designed to assess LLMs' ability to Reason with Multiple Images. This dataset encompasses a diverse range of tasks, spanning various reasoning domains such as math, physics, logic, code, table/chart understanding, and spatial and temporal reasoning. It also covers a broad spectrum of characteristics found in multi-image reasoning scenarios. We have benchmarked several cutting-edge LLMs using ReMI and found a substantial gap between their performance and human-level proficiency. This highlights the challenges in multi-image reasoning and the need for further research. Our analysis also reveals the strengths and weaknesses of different models, shedding light on the types of reasoning that are currently attainable and areas where future models require improvement. To foster further research in this area, we are releasing ReMI publicly: https://huggingface.co/datasets/mehrankazemi/ReMI.

Instructing large language models (LLMs) to solve elementary school math problems has shown great success using Chain of Thought (CoT). However, the CoT approach relies on an LLM to generate a sequence of arithmetic calculations which can be prone to cascaded calculation errors. We hypothesize that an LLM should focus on extracting predicates and generating symbolic formulas from the math problem description so that the underlying calculation can be done via an external code interpreter. We investigate using LLM to generate Prolog programs to solve mathematical questions. Experimental results show that our Prolog-based arithmetic problem-solving outperforms CoT generation in the GSM8K benchmark across three distinct LLMs. In addition, given the insensitive ordering of predicates and symbolic formulas in Prolog, we propose to permute the ground truth predicates for more robust LLM training via data augmentation.

Recent advancements in large language models (LLMs) have showcased their exceptional abilities across various tasks, such as code generation, problem-solving and reasoning. Existing benchmarks evaluate tasks in isolation, yet the extent to which LLMs can understand prose-style tasks, identify the underlying problems, and then generate appropriate code solutions is still unexplored. Addressing this gap, we introduce PECC, a novel benchmark derived from Advent Of Code (AoC) challenges and Project Euler, including 2396 problems. Unlike conventional benchmarks, PECC requires LLMs to interpret narrative-embedded problems, extract requirements, and generate executable code. A key feature of our dataset is the complexity added by natural language prompting in chat-based evaluations, mirroring real-world instruction ambiguities. Results show varying model performance between narrative and neutral problems, with specific challenges in the Euler math-based subset with GPT-3.5-Turbo passing 50% of the AoC challenges and only 8% on the Euler problems. By probing the limits of LLMs' capabilities, our benchmark provides a framework to monitor and assess the subsequent progress of LLMs as a universal problem solver.

While large language models have facilitated breakthroughs in many applications of artificial intelligence, their inherent largeness makes them computationally expensive and challenging to deploy in resource-constrained settings. In this paper, we document the development of SmolLM2, a state-of-the-art "small" (1.7 billion parameter) language model (LM). To attain strong performance, we overtrain SmolLM2 on ~11 trillion tokens of data using a multi-stage training process that mixes web text with specialized math, code, and instruction-following data. We additionally introduce new specialized datasets (FineMath, Stack-Edu, and SmolTalk) at stages where we found existing datasets to be problematically small or low-quality. To inform our design decisions, we perform both small-scale ablations as well as a manual refinement process that updates the dataset mixing rates at each stage based on the performance at the previous stage. Ultimately, we demonstrate that SmolLM2 outperforms other recent small LMs including Qwen2.5-1.5B and Llama3.2-1B. To facilitate future research on LM development as well as applications of small LMs, we release both SmolLM2 as well as all of the datasets we prepared in the course of this project.

Mathematical reasoning poses a significant challenge for language models due to its complex and structured nature. In this paper, we introduce DeepSeekMath 7B, which continues pre-training DeepSeek-Coder-Base-v1.5 7B with 120B math-related tokens sourced from Common Crawl, together with natural language and code data. DeepSeekMath 7B has achieved an impressive score of 51.7% on the competition-level MATH benchmark without relying on external toolkits and voting techniques, approaching the performance level of Gemini-Ultra and GPT-4. Self-consistency over 64 samples from DeepSeekMath 7B achieves 60.9% on MATH. The mathematical reasoning capability of DeepSeekMath is attributed to two key factors: First, we harness the significant potential of publicly available web data through a meticulously engineered data selection pipeline. Second, we introduce Group Relative Policy Optimization (GRPO), a variant of Proximal Policy Optimization (PPO), that enhances mathematical reasoning abilities while concurrently optimizing the memory usage of PPO.

Large Language Models (LLMs), despite being trained on text alone, surprisingly develop rich visual priors. These priors allow latent visual capabilities to be unlocked for vision tasks with a relatively small amount of multimodal data, and in some cases, to perform visual tasks without ever having seen an image. Through systematic analysis, we reveal that visual priors-the implicit, emergent knowledge about the visual world acquired during language pre-training-are composed of separable perception and reasoning priors with unique scaling trends and origins. We show that an LLM's latent visual reasoning ability is predominantly developed by pre-training on reasoning-centric data (e.g., code, math, academia) and scales progressively. This reasoning prior acquired from language pre-training is transferable and universally applicable to visual reasoning. In contrast, a perception prior emerges more diffusely from broad corpora, and perception ability is more sensitive to the vision encoder and visual instruction tuning data. In parallel, text describing the visual world proves crucial, though its performance impact saturates rapidly. Leveraging these insights, we propose a data-centric recipe for pre-training vision-aware LLMs and verify it in 1T token scale pre-training. Our findings are grounded in over 100 controlled experiments consuming 500,000 GPU-hours, spanning the full MLLM construction pipeline-from LLM pre-training to visual alignment and supervised multimodal fine-tuning-across five model scales, a wide range of data categories and mixtures, and multiple adaptation setups. Along with our main findings, we propose and investigate several hypotheses, and introduce the Multi-Level Existence Bench (MLE-Bench). Together, this work provides a new way of deliberately cultivating visual priors from language pre-training, paving the way for the next generation of multimodal LLMs.

GPT-3.5 and GPT-4 are the two most widely used large language model (LLM) services. However, when and how these models are updated over time is opaque. Here, we evaluate the March 2023 and June 2023 versions of GPT-3.5 and GPT-4 on four diverse tasks: 1) solving math problems, 2) answering sensitive/dangerous questions, 3) generating code and 4) visual reasoning. We find that the performance and behavior of both GPT-3.5 and GPT-4 can vary greatly over time. For example, GPT-4 (March 2023) was very good at identifying prime numbers (accuracy 97.6%) but GPT-4 (June 2023) was very poor on these same questions (accuracy 2.4%). Interestingly GPT-3.5 (June 2023) was much better than GPT-3.5 (March 2023) in this task. GPT-4 was less willing to answer sensitive questions in June than in March, and both GPT-4 and GPT-3.5 had more formatting mistakes in code generation in June than in March. Overall, our findings shows that the behavior of the same LLM service can change substantially in a relatively short amount of time, highlighting the need for continuous monitoring of LLM quality.

The pace of evolution of Large Language Models (LLMs) necessitates new approaches for rigorous and comprehensive evaluation. Traditional human annotation is increasingly impracticable due to the complexities and costs involved in generating high-quality, challenging problems. In this work, we introduce CHASE, a unified framework to synthetically generate challenging problems using LLMs without human involvement. For a given task, our approach builds a hard problem in a bottom-up manner from simpler components. Moreover, our framework decomposes the generation process into independently verifiable sub-tasks, thereby ensuring a high level of quality and correctness. We implement CHASE to create evaluation benchmarks across three diverse domains: (1) document-based question answering, (2) repository-level code completion, and (3) math reasoning. The performance of state-of-the-art LLMs on these synthetic benchmarks lies in the range of 40-60% accuracy, thereby demonstrating the effectiveness of our framework at generating challenging problems. We publicly release our benchmarks and code.

Chain-of-thought (CoT) prompting for language models demonstrates impressive performance across reasoning tasks, but typically needs labeled exemplars of the reasoning process. In this work, we introduce a new prompting approach, Analogical Prompting, designed to automatically guide the reasoning process of large language models. Inspired by analogical reasoning, a cognitive process in which humans draw from relevant past experiences to tackle new problems, our approach prompts language models to self-generate relevant exemplars or knowledge in the context, before proceeding to solve the given problem. This method presents several advantages: it obviates the need for labeling or retrieving exemplars, offering generality and convenience; it can also tailor the generated exemplars and knowledge to each problem, offering adaptability. Experimental results show that our approach outperforms 0-shot CoT and manual few-shot CoT in a variety of reasoning tasks, including math problem solving in GSM8K and MATH, code generation in Codeforces, and other reasoning tasks in BIG-Bench.

Test-Time Scaling (TTS) enhances the reasoning ability of large language models (LLMs) by allocating additional computation during inference. However, existing approaches primarily rely on output-level sampling while overlooking the role of model architecture. In mainstream Mixture-of-Experts (MoE) LLMs, we observe that varying the number of activated experts yields complementary solution sets with stable accuracy, revealing a new and underexplored source of diversity. Motivated by this observation, we propose Dynamic Experts Search (DES), a TTS strategy that elevates expert activation into a controllable dimension of the search space. DES integrates two key components: (1) Dynamic MoE, which enables direct control of expert counts during inference to generate diverse reasoning trajectories without additional cost; and (2) Expert Configuration Inheritance, which preserves consistent expert counts within a reasoning path while varying them across runs, thereby balancing stability and diversity throughout the search. Extensive experiments across MoE architectures, verifiers and reasoning benchmarks (i.e., math, code and knowledge) demonstrate that DES reliably outperforms TTS baselines, enhancing accuracy and stability without additional cost. These results highlight DES as a practical and scalable form of architecture-aware TTS, illustrating how structural flexibility in modern LLMs can advance reasoning.

Enhancing the mathematical reasoning of Large Language Models (LLMs) is a pivotal challenge in advancing AI capabilities. While Supervised Fine-Tuning (SFT) and Reinforcement Learning (RL) are the dominant training paradigms, a systematic methodology for combining them to maximize both accuracy and efficiency remains largely unexplored. This paper introduces a practical and effective training recipe that strategically integrates extended SFT with RL from online inference (GRPO). We posit that these methods play complementary, not competing, roles: a prolonged SFT phase first pushes the model's accuracy to its limits, after which a GRPO phase dramatically improves token efficiency while preserving this peak performance. Our experiments reveal that extending SFT for as many as 10 epochs is crucial for performance breakthroughs, and that the primary role of GRPO in this framework is to optimize solution length. The efficacy of our recipe is rigorously validated through top-tier performance on challenging benchmarks, including a high rank among over 2,200 teams in the strictly leak-free AI Mathematical Olympiad (AIMO). This work provides the community with a battle-tested blueprint for developing state-of-the-art mathematical reasoners that are both exceptionally accurate and practically efficient. To ensure full reproducibility and empower future research, we will open-source our entire framework, including all code, model checkpoints, and training configurations at https://github.com/analokmaus/kaggle-aimo2-fast-math-r1.

Common self-improvement approaches for large language models (LLMs), such as STaR (Zelikman et al., 2022), iteratively fine-tune LLMs on self-generated solutions to improve their problem-solving ability. However, these approaches discard the large amounts of incorrect solutions generated during this process, potentially neglecting valuable information in such solutions. To address this shortcoming, we propose V-STaR that utilizes both the correct and incorrect solutions generated during the self-improvement process to train a verifier using DPO that judges correctness of model-generated solutions. This verifier is used at inference time to select one solution among many candidate solutions. Running V-STaR for multiple iterations results in progressively better reasoners and verifiers, delivering a 4% to 17% test accuracy improvement over existing self-improvement and verification approaches on common code generation and math reasoning benchmarks with LLaMA2 models.

Recent Large Language Models (LLMs) have reported high accuracy on reasoning benchmarks. However, it is still unclear whether the observed results arise from true reasoning or from statistical recall of the training set. Inspired by the ladder of causation (Pearl, 2009) and its three levels (associations, interventions and counterfactuals), this paper introduces RE-IMAGINE, a framework to characterize a hierarchy of reasoning ability in LLMs, alongside an automated pipeline to generate problem variations at different levels of the hierarchy. By altering problems in an intermediate symbolic representation, RE-IMAGINE generates arbitrarily many problems that are not solvable using memorization alone. Moreover, the framework is general and can work across reasoning domains, including math, code, and logic. We demonstrate our framework on four widely-used benchmarks to evaluate several families of LLMs, and observe reductions in performance when the models are queried with problem variations. These assessments indicate a degree of reliance on statistical recall for past performance, and open the door to further research targeting skills across the reasoning hierarchy.

The rapid advancement of large language models (LLMs) such as GPT-3, PaLM, and Llama has significantly transformed natural language processing, showcasing remarkable capabilities in understanding and generating language. However, these models often struggle with tasks requiring complex reasoning, particularly in mathematical problem-solving, due in part to the scarcity of large-scale, high-quality, domain-specific datasets necessary for training sophisticated reasoning abilities. To address this limitation, we introduce Template-based Data Generation (TDG), a novel approach that leverages LLMs (GPT-4) to automatically generate parameterized meta-templates, which are then used to synthesize a vast array of high-quality problems and solutions. Leveraging TDG, we create TemplateMath Part I: TemplateGSM, a dataset comprising over 7 million synthetically generated grade school math problems--each accompanied by code-based and natural language solutions--with the potential to generate an effectively unlimited number more. This dataset alleviates the scarcity of large-scale mathematical datasets and serves as a valuable resource for pre-training, fine-tuning, and evaluating LLMs in mathematical reasoning. Our method not only enables the generation of virtually infinite data but also elevates data augmentation to a new level by using GPT-4 for meta-template generation, ensuring diverse and high-quality problem structures. The TemplateMath Part I: TemplateGSM dataset is publicly available at https://huggingface.co/datasets/math-ai/TemplateGSM. The code is available at https://github.com/iiis-ai/TemplateMath.

Language models typically need to be trained or finetuned in order to acquire new knowledge, which involves updating their weights. We instead envision language models that can simply read and memorize new data at inference time, thus acquiring new knowledge immediately. In this work, we extend language models with the ability to memorize the internal representations of past inputs. We demonstrate that an approximate kNN lookup into a non-differentiable memory of recent (key, value) pairs improves language modeling across various benchmarks and tasks, including generic webtext (C4), math papers (arXiv), books (PG-19), code (Github), as well as formal theorems (Isabelle). We show that the performance steadily improves when we increase the size of memory up to 262K tokens. On benchmarks including code and mathematics, we find that the model is capable of making use of newly defined functions and theorems during test time.

Model merging is an effective post-training strategy for composing capabilities in large language models without joint retraining. We study this in the supervised fine-tuning (SFT) stage, where multiple capability-based SFT checkpoints -- spanning math, code, precise instruction following, general instruction following, and knowledge recall -- must be consolidated into a single model. We introduce Optimization Trajectory Aware (OTA) Merging, a curvature-aware aggregation that leverages optimizer second-moment statistics as a diagonal curvature proxy to reweight parameter edits and mitigate interference. Complementing OTA, we propose Fast Fisher Grafting (FFG), a curvature-driven task-localization step that sparsifies conflicting or low-importance edits. FFG induces extremely low-rank masks concentrated in early attention query/key projections and token embeddings, exploiting shared curvature across capabilities. We further develop a memory-light compression of the second moments that preserves OTA's effect. Across diverse capability-based SFT checkpoints, OTA+FFG improves merged-model quality over strong weight-space baselines, reduces negative transfer, and remains robust across sparsity levels. Analyses reveal substantial curvature overlap between checkpoints, offering a novel lens on why simple linear merging can be effective in practice. Ablations confirm that FFG is critical for reducing task interference and that the compressed second moments retain the gains of the full formulation. To facilitate reproducibility, we open-source all code, training and evaluation scripts, visualization artifacts, and capability-specific SFT checkpoints at https://github.com/pmahdavi/ota-merge.

Large Language Models (LLMs) have shown great promise in tool-making, yet existing frameworks often struggle to efficiently construct reliable toolsets and are limited to single-task settings. To address these challenges, we propose GATE (Graph-based Adaptive Tool Evolution), an adaptive framework that dynamically constructs and evolves a hierarchical graph of reusable tools across multiple scenarios. We evaluate GATE on open-ended tasks (Minecraft), agent-based tasks (TextCraft, DABench), and code generation tasks (MATH, Date, TabMWP). Our results show that GATE achieves up to 4.3x faster milestone completion in Minecraft compared to the previous SOTA, and provides an average improvement of 9.23% over existing tool-making methods in code generation tasks and 10.03% in agent tasks. GATE demonstrates the power of adaptive evolution, balancing tool quantity, complexity, and functionality while maintaining high efficiency. Code and data are available at https://github.com/ayanami2003/GATE.

Language model performance depends on identifying the optimal mixture of data groups to train on (e.g., law, code, math). Prior work has proposed a diverse set of methods to efficiently learn mixture proportions, ranging from fitting regression models over training runs to dynamically updating proportions throughout training. Surprisingly, we find that no existing method consistently outperforms a simple stratified sampling baseline in terms of average test perplexity. To understand this inconsistency, we unify existing methods into a standard framework, showing they are equivalent to solving a common optimization problem: minimize average loss subject to a method-specific mixing law -- an implicit assumption on the relationship between loss and mixture proportions. This framework suggests that measuring the fidelity of a method's mixing law can offer insights into its performance. Empirically, we find that existing methods set their mixing law parameters inaccurately, resulting in the inconsistent mixing performance we observe. Using this insight, we derive a new online method named Aioli, which directly estimates the mixing law parameters throughout training and uses them to dynamically adjust proportions. Aioli outperforms stratified sampling on 6 out of 6 datasets by an average of 0.27 test perplexity points, whereas existing methods fail to consistently beat stratified sampling, doing up to 6.9 points worse. Moreover, in a practical setting where proportions are learned on shorter runs due to computational constraints, Aioli can dynamically adjust these proportions over the full training run, consistently improving performance over existing methods by up to 12.012 test perplexity points.

Recently, AI assistants based on large language models (LLMs) show surprising performance in many tasks, such as dialogue, solving math problems, writing code, and using tools. Although LLMs possess intensive world knowledge, they still make factual errors when facing some knowledge intensive tasks, like open-domain question answering. These untruthful responses from the AI assistant may cause significant risks in practical applications. We believe that an AI assistant's refusal to answer questions it does not know is a crucial method for reducing hallucinations and making the assistant truthful. Therefore, in this paper, we ask the question "Can AI assistants know what they don't know and express them through natural language?" To answer this question, we construct a model-specific "I don't know" (Idk) dataset for an assistant, which contains its known and unknown questions, based on existing open-domain question answering datasets. Then we align the assistant with its corresponding Idk dataset and observe whether it can refuse to answer its unknown questions after alignment. Experimental results show that after alignment with Idk datasets, the assistant can refuse to answer most its unknown questions. For questions they attempt to answer, the accuracy is significantly higher than before the alignment.

The recently released Google Gemini class of models are the first to comprehensively report results that rival the OpenAI GPT series across a wide variety of tasks. In this paper, we do an in-depth exploration of Gemini's language abilities, making two contributions. First, we provide a third-party, objective comparison of the abilities of the OpenAI GPT and Google Gemini models with reproducible code and fully transparent results. Second, we take a closer look at the results, identifying areas where one of the two model classes excels. We perform this analysis over 10 datasets testing a variety of language abilities, including reasoning, answering knowledge-based questions, solving math problems, translating between languages, generating code, and acting as instruction-following agents. From this analysis, we find that Gemini Pro achieves accuracy that is close but slightly inferior to the corresponding GPT 3.5 Turbo on all tasks that we benchmarked. We further provide explanations for some of this under-performance, including failures in mathematical reasoning with many digits, sensitivity to multiple-choice answer ordering, aggressive content filtering, and others. We also identify areas where Gemini demonstrates comparably high performance, including generation into non-English languages, and handling longer and more complex reasoning chains. Code and data for reproduction can be found at https://github.com/neulab/gemini-benchmark

We study the process through which reasoning models trained with reinforcement learning on verifiable rewards (RLVR) can learn to solve new problems. We find that RLVR drives performance in two main ways: (1) by compressing pass@k into pass@1 and (2) via "capability gain" in which models learn to solve new problems that they previously could not solve even at high k. We find that while capability gain exists across model scales, learning to solve new problems is primarily driven through self-distillation. We demonstrate these findings across model scales ranging from 0.5B to 72B parameters on >500,000 reasoning problems with prompts and verifiable final answers across math, science, and code domains. We further show that we can significantly improve pass@k rates by leveraging natural language guidance for the model to consider within context while still requiring the model to derive a solution chain from scratch. Based of these insights, we derive Guide -- a new class of online training algorithms. Guide adaptively incorporates hints into the model's context on problems for which all rollouts were initially incorrect and adjusts the importance sampling ratio for the "off-policy" trajectories in order to optimize the policy for contexts in which the hints are no longer present. We describe variants of Guide for GRPO and PPO and empirically show that Guide-GRPO on 7B and 32B parameter models improves generalization over its vanilla counterpart with up to 4% macro-average improvement across math benchmarks. We include careful ablations to analyze Guide's components and theoretically analyze Guide's learning efficiency.

Inference-time scaling via repeated sampling has shown promise in reasoning tasks, but its effectiveness in multilingual generation remains underexplored. We evaluate this approach using perplexity- and reward-based verifiers on two multilingual benchmarks: the Aya Evaluation Suite and m-ArenaHard. Our results show consistent quality improvements, with gains exceeding 35% in some cases. While perplexity-based scoring is effective for open-ended prompts, only reward-based verifiers improve performance on tasks requiring reasoning (e.g., math, code). Our results demonstrate the broader utility of repeated sampling for multilingual text generation and underscore the importance of selecting right verifiers for the task.

Recently, inspired by the concept of sparsity, Mixture-of-Experts (MoE) models have gained increasing popularity for scaling model size while keeping the number of activated parameters constant. In this study, we thoroughly investigate the sparsity of the dense LLaMA model by constructing MoE for both the attention (i.e., Attention MoE) and MLP (i.e., MLP MoE) modules in the transformer blocks. Specifically, we investigate different expert construction methods and granularities under the same activation conditions to analyze the impact of sparsifying the model. Additionally, to comprehensively evaluate the model's capabilities across various domains (e.g., conversation, code, math) after sparsification, we apply sparsity to the instructed large language models (LLMs) and construct instructed MoE models. To counteract the performance degradation resulting from increased sparsity, we design a two-stage post-training strategy to enhance model performance. Experiments on the LLaMA3 model demonstrate the potential effectiveness of this approach for future developments of instructed MoE models. The source codes and models are available at: https://github.com/OpenSparseLLMs/LLaMA-MoE-v2.

In this report, we introduce Qwen2.5, a comprehensive series of large language models (LLMs) designed to meet diverse needs. Compared to previous iterations, Qwen 2.5 has been significantly improved during both the pre-training and post-training stages. In terms of pre-training, we have scaled the high-quality pre-training datasets from the previous 7 trillion tokens to 18 trillion tokens. This provides a strong foundation for common sense, expert knowledge, and reasoning capabilities. In terms of post-training, we implement intricate supervised finetuning with over 1 million samples, as well as multistage reinforcement learning. Post-training techniques enhance human preference, and notably improve long text generation, structural data analysis, and instruction following. To handle diverse and varied use cases effectively, we present Qwen2.5 LLM series in rich sizes. Open-weight offerings include base and instruction-tuned models, with quantized versions available. In addition, for hosted solutions, the proprietary models currently include two mixture-of-experts (MoE) variants: Qwen2.5-Turbo and Qwen2.5-Plus, both available from Alibaba Cloud Model Studio. Qwen2.5 has demonstrated top-tier performance on a wide range of benchmarks evaluating language understanding, reasoning, mathematics, coding, human preference alignment, etc. Specifically, the open-weight flagship Qwen2.5-72B-Instruct outperforms a number of open and proprietary models and demonstrates competitive performance to the state-of-the-art open-weight model, Llama-3-405B-Instruct, which is around 5 times larger. Qwen2.5-Turbo and Qwen2.5-Plus offer superior cost-effectiveness while performing competitively against GPT-4o-mini and GPT-4o respectively. Additionally, as the foundation, Qwen2.5 models have been instrumental in training specialized models such as Qwen2.5-Math, Qwen2.5-Coder, QwQ, and multimodal models.

Language model pretraining with next token prediction has proved effective for scaling compute but is limited to the amount of available training data. Scaling reinforcement learning (RL) unlocks a new axis for the continued improvement of artificial intelligence, with the promise that large language models (LLMs) can scale their training data by learning to explore with rewards. However, prior published work has not produced competitive results. In light of this, we report on the training practice of Kimi k1.5, our latest multi-modal LLM trained with RL, including its RL training techniques, multi-modal data recipes, and infrastructure optimization. Long context scaling and improved policy optimization methods are key ingredients of our approach, which establishes a simplistic, effective RL framework without relying on more complex techniques such as Monte Carlo tree search, value functions, and process reward models. Notably, our system achieves state-of-the-art reasoning performance across multiple benchmarks and modalities -- e.g., 77.5 on AIME, 96.2 on MATH 500, 94-th percentile on Codeforces, 74.9 on MathVista -- matching OpenAI's o1. Moreover, we present effective long2short methods that use long-CoT techniques to improve short-CoT models, yielding state-of-the-art short-CoT reasoning results -- e.g., 60.8 on AIME, 94.6 on MATH500, 47.3 on LiveCodeBench -- outperforming existing short-CoT models such as GPT-4o and Claude Sonnet 3.5 by a large margin (up to +550%).

Aligning large language models (LLMs) with human preferences has achieved remarkable success. However, existing Chinese preference datasets are limited by small scale, narrow domain coverage, and lack of rigorous data validation. Additionally, the reliance on human annotators for instruction and response labeling significantly constrains the scalability of human preference datasets. To address these challenges, we design an LLM-based Chinese preference dataset annotation pipeline with no human intervention. Specifically, we crawled and carefully filtered 92k high-quality Chinese queries and employed 15 mainstream LLMs to generate and score chosen-rejected response pairs. Based on it, we introduce COIG-P (Chinese Open Instruction Generalist - Preference), a high-quality, large-scale Chinese preference dataset, comprises 1,009k Chinese preference pairs spanning 6 diverse domains: Chat, Code, Math, Logic, Novel, and Role. Building upon COIG-P, to reduce the overhead of using LLMs for scoring, we trained a 8B-sized Chinese Reward Model (CRM) and meticulously constructed a Chinese Reward Benchmark (CRBench). Evaluation results based on AlignBench liu2024alignbenchbenchmarkingchinesealignment show that that COIG-P significantly outperforms other Chinese preference datasets, and it brings significant performance improvements ranging from 2% to 12% for the Qwen2/2.5 and Infinity-Instruct-3M-0625 model series, respectively. The results on CRBench demonstrate that our CRM has a strong and robust scoring ability. We apply it to filter chosen-rejected response pairs in a test split of COIG-P, and our experiments show that it is comparable to GPT-4o in identifying low-quality samples while maintaining efficiency and cost-effectiveness. Our codes and data are released in https://github.com/multimodal-art-projection/COIG-P.

Recently, o1-like models have drawn significant attention, where these models produce the long Chain-of-Thought (CoT) reasoning steps to improve the reasoning abilities of existing Large Language Models (LLMs). In this paper, to understand the qualities of these long CoTs and measure the critique abilities of existing LLMs on these long CoTs, we introduce the DeltaBench, including the generated long CoTs from different o1-like models (e.g., QwQ, DeepSeek-R1) for different reasoning tasks (e.g., Math, Code, General Reasoning), to measure the ability to detect errors in long CoT reasoning. Based on DeltaBench, we first perform fine-grained analysis of the generated long CoTs to discover the effectiveness and efficiency of different o1-like models. Then, we conduct extensive evaluations of existing process reward models (PRMs) and critic models to detect the errors of each annotated process, which aims to investigate the boundaries and limitations of existing PRMs and critic models. Finally, we hope that DeltaBench could guide developers to better understand the long CoT reasoning abilities of their models.

Coordinating multiple large language models (LLMs) to solve complex tasks collaboratively poses a fundamental trade-off between the computation costs and collective performance compared with individual model. We introduce a novel, game-theoretically grounded reinforcement learning (RL) framework, the Multi-Agent Cooperation Sequential Public Goods Game (MAC-SPGG), to systematically incentivize cooperation in multi-LLM ensembles. In MAC-SPGG, LLM agents move in sequence, observing predecessors' outputs and updating beliefs to condition their own contributions. By redesigning the public-goods reward, effortful contributions become the unique Subgame Perfect Nash Equilibrium (SPNE), which eliminates free-riding under traditional SPGG or PGG. Its sequential protocol replaces costly round-based information exchanges with a streamlined decision flow, cutting communication overhead while retaining strategic depth. We prove the existence and uniqueness of the SPNE under realistic parameters, and empirically show that MAC-SPGG-trained ensembles outperform single-agent baselines, chain-of-thought prompting, and other cooperative methods, even achieving comparable performance to large-scale models across reasoning, math, code generation, and NLP tasks. Our results highlight the power of structured, incentive-aligned MAC-SPGG cooperation for scalable and robust multi-agent language generation.

Inference-time techniques are emerging as highly effective tools to enhance large language model (LLM) capabilities. However, best practices for developing systems that combine these techniques remain underdeveloped due to our limited understanding of the utility of individual inference-time techniques and the interactions between them. Additionally, efficiently and automatically searching the space of model choices, inference-time techniques, and their compositions is challenging due to the large design space. To address these challenges, we introduce Archon, a modular framework for selecting, combining, and stacking layers of inference-time techniques to construct optimized LLM systems for target benchmarks. Rather than relying on a single LLM called once, we leverage a diverse set of LLMs and inference-time techniques, creating LLM systems greater than the sum of their parts. Archon defines an extensible design space, encompassing techniques such as generation ensembling, repeated sampling, ranking, fusion, critiquing, verification, and unit testing. It transforms the problem of building LLM systems into a hyperparameter optimization objective. Given the available LLMs, inference-time techniques, and compute budget, Archon utilizes hyperparameter search techniques to discover optimized architectures for target benchmark(s). We evaluate Archon architectures across a range of instruction-following, reasoning, and coding benchmarks, including MT-Bench, Arena-Hard-Auto, AlpacaEval 2.0, MixEval, MixEval Hard, MATH, and CodeContests. Archon architectures outperform frontier models, such as GPT-4o and Claude 3.5 Sonnet, on these benchmarks, achieving an average accuracy increase of 15.1 percentage points by using all available LLMs. We make our code and datasets available publicly on Github: https://github.com/ScalingIntelligence/Archon.