Daily Papers

Large reasoning models have recently made significant strides in mathematical and code reasoning, yet their success has not transferred smoothly to the medical domain. While multiple factors contribute to this disparity, a critical issue is the inadequate focus on the quality of intermediate reflection steps, which is particularly crucial in high-stakes medical scenarios. To address this challenge, we propose Med-REFL, a \textbf{Med}ical \textbf{R}easoning \textbf{E}nhancement via self-corrected \textbf{F}ine-grained ref\textbf{L}ection. Our method leverages a tree-of-thought approach to decompose medical questions into fine-grained reasoning paths, quantitatively evaluating each step and its subsequent reflections. These assessments enable automatic construction of direct preference optimization data, reducing reliance on expensive expert annotations while guiding models to identify and correct reasoning errors. Experimental results on the MedQA-USMLE benchmark demonstrate Med-REFL achieves consistent improvements, with average gains up to 4.11\%. Notably, it further boosts the state-of-the-art performance of 7B/8B models by an additional 4.13\%. Furthermore, Med-REFL exhibits strong generalization capabilities and robustness across several challenging medical question-answering datasets. Our work illustrates that prioritizing reflection quality leads to more accurate and trustworthy reasoning in medical AI applications. Checkpoints, code, and data can be found https://github.com/TianYin123/Med-REFL{here}.

Large Reasoning Models (LRMs) have recently achieved strong mathematical and code reasoning performance through Reinforcement Learning (RL) post-training. However, we show that modern reasoning post-training induces an unintended exploration collapse: temperature-based sampling no longer increases pass@n accuracy. Empirically, the final-layer posterior of post-trained LRMs exhibit sharply reduced entropy, while the entropy of intermediate layers remains relatively high. Motivated by this entropy asymmetry, we propose Latent Exploration Decoding (LED), a depth-conditioned decoding strategy. LED aggregates intermediate posteriors via cumulative sum and selects depth configurations with maximal entropy as exploration candidates. Without additional training or parameters, LED consistently improves pass@1 and pass@16 accuracy by 0.61 and 1.03 percentage points across multiple reasoning benchmarks and models. Project page: https://GitHub.com/Xiaomi-Research/LED.

Quantization-aware distillation (QAD) restores much of the short-form question-answering performance lost to sub-3-bit quantization, yet leaves mathematical and code reasoning substantially impaired. Long generations often degenerate into repetitive loops, exhausting the decoding budget without completing a solution. We trace this gap to quantization-amplified exposure bias: QAD trains on fixed corpus prefixes, while quantization-induced deviations compound along the model's own autoregressive trajectories. To address this mismatch, we introduce an on-policy distillation (OPD) stage that places teacher supervision where the quantized model actually goes. Starting from a QAD checkpoint, the student generates through the quantized forward path used at deployment and receives feedback from a frozen full-precision teacher on its own prefixes, combining dense token-level guidance with task-verifier rewards. Across four models at 2.79 and 1.88 effective bits, OPD raises average BF16 performance retention from 35% to 70% on MATH-500 and from 66% to 91% on HumanEval while preserving short-form performance, with reasoning gains substantially exceeding those of continued teacher-forced QAD in matched-budget comparisons. By coupling QAD's stable low-bit initialization with OPD's on-policy reasoning recovery, our framework provides a comprehensive sub-3-bit solution that preserves broad capabilities while restoring long-form reasoning.

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.

While plan-and-infill decoding in Masked Diffusion Models (MDMs) shows promise for mathematical and code reasoning, performance remains highly sensitive to slot infilling order, often yielding substantial output variance. We introduce McDiffuSE, a framework that formulates slot selection as decision making and optimises infilling orders through Monte Carlo Tree Search (MCTS). McDiffuSE uses look-ahead simulations to evaluate partial completions before commitment, systematically exploring the combinatorial space of generation orders. Experiments show an average improvement of 3.2% over autoregressive baselines and 8.0% over baseline plan-and-infill, with notable gains of 19.5% on MBPP and 4.9% on MATH500. Our analysis reveals that while McDiffuSE predominantly follows sequential ordering, incorporating non-sequential generation is essential for maximising performance. We observe that larger exploration constants, rather than increased simulations, are necessary to overcome model confidence biases and discover effective orderings. These findings establish MCTS-based planning as an effective approach for enhancing generation quality in MDMs.

Policy-shift distillation reuses the difference between a teacher's log probabilities before and after reinforcement learning (RL) as supervision on student-generated trajectories. However, this endpoint policy shift cannot reveal whether local action preferences evolved consistently during teacher training. The same endpoint shift can result from early and late shifts that reinforce or partially cancel each other. We investigate whether agreement between early and late teacher policy shifts can guide selective transfer. We introduce Waypoint-guided On-Policy Distillation (Waypoint-OPD), which uses an intermediate teacher checkpoint from the same RL run to compare early and late shifts in relative action preferences at student-visited prefixes. We gate the endpoint shift according to the directional agreement between the early and late shifts, suppressing it when they point in opposing directions. Compared with endpoint-only transfer, our method requires only one additional frozen-teacher scoring pass. Using RL-trained Qwen3-4B teachers, we evaluate our method with Qwen3-8B, Qwen3-14B, and Qwen3-30B-A3B as student models in mathematical and code reasoning. In all six domain--student settings, Waypoint-OPD achieves the highest domain-average score among the compared distillation methods. Ablations with the Qwen3-8B student show that removing or shuffling gates reduces mathematical reasoning performance. These results support the use of directional agreement between early and late teacher policy shifts to weight the endpoint shift during student training.

Mathematical reasoning remains one of the most challenging domains for large language models (LLMs), requiring not only linguistic understanding but also structured logical deduction and numerical precision. While recent LLMs demonstrate strong general-purpose reasoning abilities, their mathematical competence across diverse languages remains underexplored. Existing benchmarks primarily focus on English or a narrow subset of high-resource languages, leaving significant gaps in assessing multilingual and cross-lingual mathematical reasoning. To address this, we introduce MathMist, a parallel multilingual benchmark for mathematical problem solving and reasoning. MathMist encompasses over 21K aligned question-answer pairs across seven languages, representing a balanced coverage of high-, medium-, and low-resource linguistic settings. The dataset captures linguistic variety, multiple types of problem settings, and solution synthesizing capabilities. We systematically evaluate a diverse suite of models, including open-source small and medium LLMs, proprietary systems, and multilingual-reasoning-focused models, under zero-shot, chain-of-thought (CoT), and code-switched reasoning paradigms. Our results reveal persistent deficiencies in LLMs' ability to perform consistent and interpretable mathematical reasoning across languages, with pronounced degradation in low-resource settings. All the codes and data are available at GitHub: https://github.com/mahbubhimel/MathMist

Large Language Models (LLMs) have made remarkable progress in mathematical reasoning, but still continue to struggle with high-precision tasks like numerical computation and formal symbolic manipulation. Integrating external tools has emerged as a promising approach to bridge this gap. Despite recent advances, existing methods struggle with three key challenges: constructing tool-integrated reasoning data, performing fine-grained optimization, and enhancing inference. To overcome these limitations, we propose THOR (Tool-Integrated Hierarchical Optimization via RL). First, we introduce TIRGen, a multi-agent actor-critic-based pipeline for constructing high-quality datasets of tool-integrated reasoning paths, aligning with the policy and generalizing well across diverse models. Second, to perform fine-grained hierarchical optimization, we introduce an RL strategy that jointly optimizes for both trajectory-level problem solving and step-level code generation. This is motivated by our key insight that the success of an intermediate tool call is a strong predictor of the final answer's correctness. Finally, THOR incorporates a self-correction mechanism that leverages immediate tool feedback to dynamically revise erroneous reasoning paths during inference. Our approach demonstrates strong generalization across diverse models, performing effectively in both reasoning and non-reasoning models. It further achieves state-of-the-art performance for models of a similar scale on multiple mathematical benchmarks, while also delivering consistent improvements on code benchmarks. Our code will be publicly available at https://github.com/JingMog/THOR.

Recent progress in large language models (LLMs) highlights the power of scaling test-time compute to achieve strong performance on complex tasks, such as mathematical reasoning and code generation. This raises a critical question: how should model training be modified to optimize performance under a subsequent test-time compute strategy and budget? To explore this, we focus on pass@N, a simple test-time strategy that searches for a correct answer in N independent samples. We show, surprisingly, that training with cross-entropy (CE) loss can be {it misaligned} with pass@N in that pass@N accuracy {it decreases} with longer training. We explain the origins of this misalignment in terms of model overconfidence induced by CE, and experimentally verify our prediction of overconfidence as an impediment to scaling test-time compute via pass@N. Furthermore we suggest a principled, modified training loss that is better aligned to pass@N by limiting model confidence and rescuing pass@N test performance. Our algorithm demonstrates improved mathematical reasoning on MATH and MiniF2F benchmarks under several scenarios: (1) providing answers to math questions; and (2) proving theorems by searching over proof trees of varying shapes. Overall our work underscores the importance of co-designing two traditionally separate phases of LLM development: training-time protocols and test-time search and reasoning strategies.

The role of reinforcement learning (RL) in enhancing the reasoning of large language models (LLMs) is becoming increasingly significant. Despite the success of RL in many scenarios, there are still many challenges in improving the reasoning of LLMs. One challenge is the sparse reward, which makes optimization difficult for RL and necessitates a large amount of data samples. Another challenge stems from the inherent instability of RL, particularly when using Actor-Critic (AC) methods to derive optimal policies, which often leads to unstable training processes. To address these issues, we introduce Direct Advantage Policy Optimization (DAPO), an novel step-level offline RL algorithm. Unlike standard alignment that rely solely outcome rewards to optimize policies (such as DPO), DAPO employs a critic function to predict the reasoning accuracy at each step, thereby generating dense signals to refine the generation strategy. Additionally, the Actor and Critic components in DAPO are trained independently, avoiding the co-training instability observed in standard AC algorithms like PPO. We train DAPO on mathematical and code query datasets and then evaluate its performance on multiple benchmarks. Our results show that DAPO can effectively enhance the mathematical and code capabilities on both SFT models and RL models, demonstrating the effectiveness of DAPO.

Large Language Models (LLMs) have achieved significant advances in reasoning tasks. A key approach is tree-based search with verifiers, which expand candidate reasoning paths and use reward models to guide pruning and selection. Although effective in improving accuracy, these methods are not optimal in terms of efficiency: they perform simple decomposition on the reasoning process, but ignore the planning-execution nature of tasks such as math reasoning or code generation. This results in inefficient exploration of reasoning process. To address this, we propose a dual-phase test-time scaling framework that explicitly separates reasoning into planning and execution, and performs search over the two phases individually. Specifically, we decompose reasoning trajectories and develop reward models for each phase, enabling the search to explore and prune plans and executions separately. We further introduce a dynamic budget allocation mechanism that adaptively redistributes sampling effort based on reward feedback, allowing early stopping on confident steps and reallocation of computation to more challenging parts of the reasoning process. Experiments on both mathematical reasoning and code generation benchmarks demonstrate that our approach consistently improves accuracy while reducing redundant computation.

Inference-time compute has re-emerged as a practical way to improve LLM reasoning. Most test-time scaling (TTS) algorithms rely on autoregressive decoding, which is ill-suited to discrete diffusion language models (dLLMs) due to their parallel decoding over the entire sequence. As a result, developing effective and efficient TTS methods to unlock dLLMs' full generative potential remains an underexplored challenge. To address this, we propose Prism (Pruning, Remasking, and Integrated Self-verification Method), an efficient TTS framework for dLLMs that (i) performs Hierarchical Trajectory Search (HTS) which dynamically prunes and reallocates compute in an early-to-mid denoising window, (ii) introduces Local branching with partial remasking to explore diverse implementations while preserving high-confidence tokens, and (iii) replaces external verifiers with Self-Verified Feedback (SVF) obtained via self-evaluation prompts on intermediate completions. Across four mathematical reasoning and code generation benchmarks on three dLLMs, including LLaDA 8B Instruct, Dream 7B Instruct, and LLaDA 2.0-mini, our Prism achieves a favorable performance-efficiency trade-off, matching best-of-N performance with substantially fewer function evaluations (NFE). The code is released at https://github.com/viiika/Prism.

Large language models (LLMs), when guided by explicit textual plans, can perform reliable step-by-step reasoning during problem-solving. However, generating accurate and effective textual plans remains challenging due to LLM hallucinations and the high diversity of task-specific questions. To address this, we draw inspiration from human Implicit Cognition (IC), the subconscious process by which decisions are guided by compact, generalized patterns learned from past experiences without requiring explicit verbalization. We propose iCLP, a novel framework that enables LLMs to adaptively generate latent plans (LPs), which are compact encodings of effective reasoning instructions. iCLP first distills explicit plans from existing step-by-step reasoning trajectories. It then learns discrete representations of these plans via a vector-quantized autoencoder coupled with a codebook. Finally, by fine-tuning LLMs on paired latent plans and corresponding reasoning steps, the models learn to perform implicit planning during reasoning. Experimental results on mathematical reasoning and code generation tasks demonstrate that, with iCLP, LLMs can plan in latent space while reasoning in language space. This approach yields significant improvements in both accuracy and efficiency and, crucially, demonstrates strong cross-domain generalization while preserving the interpretability of chain-of-thought reasoning.

On-policy distillation (OPD) has emerged as an efficient post-training paradigm for large language models. However, standard OPD requires a live teacher inference server throughout training, resulting in substantial infrastructure overhead. In this work, we investigate whether on-policy distillation can be performed offline. A natural approach is to precompute teacher log-probabilities once over SFT rollouts and reuse them during training. In practice, however, this offline variant fails to reliably match the performance of standard OPD. To understand this discrepancy, we identify a previously overlooked condition that is critical for any OPD pipeline, which we term teacher consistency. This condition requires that the same teacher model be used for both supervised fine-tuning and OPD. We show that violating teacher consistency introduces an irreducible gradient bias, causing both offline and online OPD to converge to a suboptimal fixed point regardless of training duration. Building on this insight, we propose Lightning OPD, an offline on-policy distillation framework that enforces teacher consistency by precomputing teacher log-probabilities over SFT rollouts. This design eliminates the need for a live teacher server entirely. We further show that, under teacher consistency, Lightning OPD shares the same optimum as standard OPD, with bounded gradient discrepancy and an implicit regularization effect that helps prevent policy drift. Extensive experiments on mathematical reasoning and code generation demonstrate that Lightning OPD achieves state-of-the-art performance with significantly improved efficiency. Starting from an SFT-initialized Qwen3-8B-Base model, Lightning OPD reaches 69.9% on AIME 2024 in just 30 GPU hours, achieving a 4.0x speedup over standard OPD and substantially lowering the barrier to entry for academic research on LLM post-training.

Diffusion Large Language Models (dLLMs) have recently emerged as a promising alternative to autoregressive LLMs by enabling non-autoregressive text generation. However, their practical deployment remains limited by inefficient inference, largely due to the absence of effective Key-Value (KV) caching and scalable parallel decoding mechanisms. Existing acceleration methods typically study KV caching and parallel decoding in isolation, overlooking the I/O bottlenecks that arise when cache reuse and parallel token verification are jointly applied. In this work, we introduce Flash-dLLM, a training-free inference acceleration framework for fast and memory-efficient dLLMs. Flash-dLLM first identifies GPU memory I/O as a dominant bottleneck in KV-cache-enabled dLLM inference and addresses it with an I/O-aware fused KV-cache kernel that reduces redundant memory movement. Building on this optimized cache mechanism, Flash-dLLM further proposes an efficient KV-cache-driven draft-and-verify decoding strategy, where the dLLM itself serves as both drafter and verifier without requiring an auxiliary model. This unified design enables faster decoding while preserving generation quality and improving scalability to longer sequences and larger batch size. Extensive experiments on mathematical reasoning and code-generation benchmarks demonstrate that Flash-dLLM consistently outperforms existing state-of-the-art dLLM acceleration methods in both inference speed and memory efficiency. In particular, it achieves 5.1times and 11.0times speedups over prior strongest baseline Elastic-Cache on GSM8K and HumanEval, respectively.

Reinforcement Learning with Verifiable Rewards (RLVR) has become an effective post-training method for improving the reasoning abilities of Large Language Models (LLMs), mainly by shaping higher-order behaviors such as reflection and planning. However, previous RLVR algorithms often apply uniform training signals to all tokens, without considering the different roles of low-entropy knowledge-related tokens and high-entropy reasoning-related tokens. Some recent methods try to separate these token types by gradient masking or asynchronous updates, but these approaches may break semantic dependencies in the model output and hinder effective learning. In this work, we propose Archer, an entropy-aware RLVR approach with dual-token constraints and synchronous updates. Specifically, our method applies weaker KL regularization and higher clipping thresholds to reasoning tokens to encourage exploration, while using stronger constraints on knowledge tokens to maintain factual knowledge. Experimental results on several mathematical reasoning and code generation benchmarks show that our approach significantly outperforms previous RLVR methods, reaching or exceeding state-of-the-art performance among models of comparable size. The code is available at https://github.com/wizard-III/ArcherCodeR.

Distilling the reasoning capabilities of large language models (LLMs) into smaller students is a central challenge for efficient deployment. Current approaches face a fundamental tension: optimizing purely for verifiable task rewards (e.g., via GRPO) leads to reward hacking, where students arrive at correct final answers through flawed intermediate logic, while regularizing with soft divergence penalties against a teacher (e.g., KL-based distillation) dilutes task performance and, critically, allows the student to compensate for severe logical violations at one step with high teacher agreement at others. We argue that this averaging is fundamentally misaligned with the nature of reasoning: a chain-of-thought is only as valid as its weakest link. Motivated by this observation, we formulate reasoning distillation as a constrained reinforcement learning problem in which the task reward is maximized subject to a worst-case constraint on the teacher log-likelihood along every prefix of the trajectory. To avoid the prohibitive cost of dual Lagrangian solvers and the test-time teacher dependence of state-augmented methods such as Saute, we derive an unaugmented constrained MDP whose reward transformation preserves the hard-constraint semantics, admits a low-variance policy gradient decomposition into single-step and long-term terms, and provably satisfies the worst-case constraint almost surely in the penalty limit. Through extensive experiments on mathematical reasoning and code generation tasks, we demonstrate that our method significantly expands the accuracy-fidelity Pareto front. By matching the high Final Answer Correctness of pure RL and drastically reducing teacher constraint violations, we ultimately achieve the highest rigorous Reasoning Success Rate across all evaluated settings.

Current approaches for training Process Reward Models (PRMs) often involve breaking down responses into multiple reasoning steps using rule-based techniques, such as using predefined placeholder tokens or setting the reasoning step's length into a fixed size. These approaches overlook the fact that specific words do not typically mark true decision points in a text. To address this, we propose AdaptiveStep, a method that divides reasoning steps based on the model's confidence in predicting the next word. This division method provides more decision-making information at each step, enhancing downstream tasks, such as reward model learning. Moreover, our method does not require manual annotation. We demonstrate its effectiveness through experiments with AdaptiveStep-trained PRMs in mathematical reasoning and code generation tasks. Experimental results indicate that the outcome PRM achieves state-of-the-art Best-of-N performance, surpassing greedy search strategy with token-level value-guided decoding, while also reducing construction costs by over 30% compared to existing open-source PRMs. In addition, we provide a thorough analysis and case study on the PRM's performance, transferability, and generalization capabilities.

Fully continuous diffusion language models (dLMs) denoise continuous representations without intermediate discretization, then decode all response tokens in parallel at the final step. Their performance on challenging reasoning tasks remains less established than that of autoregressive (AR) LLMs and masked dLMs. We scale Embedded Language Flows (ELF) to mathematical reasoning and code generation on GSM8K, MATH-500, HumanEval, and MBPP. We introduce ELF-REG, which improves learning with representation alignment and entanglement (REPA+REG), where a frozen AR teacher supervises intermediate denoiser features and supplies a global representation that is jointly denoised with the response. ELF-REG-L achieves 55.96% pass@1 on GSM8K at 64 network function evaluations (NFE), and 13.39% on MATH-500 and 22.56% on HumanEval at 128 NFE. It outperforms the evaluated comparable-scale dLMs in pass@1 on GSM8K and code, and improves MATH-500 pass@1 from 10.55% for the ELF-L baseline to 13.39% with ELF-REG-L. Without few-step training, the same task-specific checkpoints support strong low-NFE performance through early-stop, which decodes an intermediate clean prediction without completing the denoising trajectory. At 16 NFE, ELF-REG-L reaches 41.21% HumanEval pass@10, outperforming recent continuous dLMs of comparable scale.

In the realm of embodied artificial intelligence, the reasoning capabilities of Large Language Models (LLMs) play a pivotal role. Although there are effective methods like program-of-thought prompting for LLMs which uses programming language to tackle complex reasoning tasks, the specific impact of code data on the improvement of reasoning capabilities remains under-explored. To address this gap, we propose complexity-impacted reasoning score (CIRS), which combines structural and logical attributes, to measure the correlation between code and reasoning abilities. Specifically, we use the abstract syntax tree to encode the structural information and calculate logical complexity by considering the difficulty and the cyclomatic complexity. Through an empirical analysis, we find not all code data of complexity can be learned or understood by LLMs. Optimal level of complexity is critical to the improvement of reasoning abilities by program-aided prompting. Then we design an auto-synthesizing and stratifying algorithm, and apply it to instruction generation for mathematical reasoning and code data filtering for code generation tasks. Extensive results demonstrates the effectiveness of our proposed approach. Code will be integrated into the EasyInstruct framework at https://github.com/zjunlp/EasyInstruct.

Enhancing the reasoning capabilities of Large Language Models (LLMs) with efficiency and scalability remains a fundamental challenge in artificial intelligence research. This paper presents a rigorous experimental investigation into how difficulty-aware staged reinforcement learning (RL) strategies can substantially improve LLM reasoning performance. Through systematic analysis, we demonstrate that strategically selecting training data according to well-defined difficulty levels markedly enhances RL optimization. Moreover, we introduce a staged training methodology, progressively exposing models to increasingly challenging tasks, further amplifying reasoning capabilities. Our findings reveal significant cross-domain benefits when simultaneously training models on mathematical reasoning and code generation tasks. Notably, our proposed approach enables a 1.5B parameter model to achieve an accuracy of 42.3\% on the AIME-2024 benchmark, 89.5\% on the MATH-500 benchmark. These results underscore the efficacy of our method in advancing the reasoning proficiency of LLMs. We will open-source our datasets on GitHub and Hugging Face.

Fine-tuning large language models (LLMs) using standard first-order (FO) optimization often drives training toward sharp, poorly generalizing minima. Conversely, zeroth-order (ZO) methods offer stronger exploratory behavior without relying on explicit gradients, yet suffer from slow convergence. More critically, our analysis reveals that in generative tasks, the vast output and search space significantly amplify estimation variance, rendering ZO methods both noisy and inefficient. To address these challenges, we propose Hi-ZFO (Hierarchical Zeroth- and First-Order optimization), a hybrid framework designed to synergize the precision of FO gradients with the exploratory capability of ZO estimation. Hi-ZFO adaptively partitions the model through layer-wise importance profiling, applying precise FO updates to critical layers while leveraging ZO optimization for less sensitive ones. Notably, ZO in Hi-ZFO is not merely a memory-saving surrogate; it is intentionally introduced as a source of "beneficial stochasticity" to help the model escape the local minima where pure FO optimization tends to stagnate. Validated across diverse generative, mathematical, and code reasoning tasks, Hi-ZFO consistently achieves superior performance while significantly reducing the training time. These results demonstrate the effectiveness of hierarchical hybrid optimization for LLM fine-tuning.

Recent advancements in large language models (LLMs) underscore the need for stronger reasoning capabilities to solve complex problems effectively. While Chain-of-Thought (CoT) reasoning has been a step forward, it remains insufficient for many domains. A promising alternative is explicit high-level plan generation, but existing approaches largely assume that LLMs can produce effective plans through few-shot prompting alone, without additional training. In this work, we challenge this assumption and introduce CRISP (Complex Reasoning with Interpretable Step-based Plans), a multi-domain dataset of high-level plans for mathematical reasoning and code generation. The plans in CRISP are automatically generated and rigorously validated--both intrinsically, using an LLM as a judge, and extrinsically, by evaluating their impact on downstream task performance. We demonstrate that fine-tuning a small model on CRISP enables it to generate higher-quality plans than much larger models using few-shot prompting, while significantly outperforming Chain-of-Thought reasoning. Furthermore, our out-of-domain evaluation reveals that fine-tuning on one domain improves plan generation in the other, highlighting the generalizability of learned planning capabilities.

Discrete diffusion language models (dLLMs) generate text by iteratively denoising a masked sequence. However, standard dLLMs condition each denoising step solely on the current hard-masked sequence, while intermediate continuous representations are discarded after sampling and remasking. We term this bottleneck the Information Island issue: continuous information remains isolated within individual denoising steps and fails to propagate across the trajectory. This bottleneck is especially harmful for reasoning, which requires intermediate reasoning state to be preserved and updated across many denoising steps. To address this limitation, we introduce MetaState, a lightweight recurrent augmentation that equips a frozen dLLM backbone with persistent, fixed-size working memory. MetaState comprises three modules with a shared time conditioner: a cross-attention Mixer that reads backbone activations into memory slots, a GRU-style Updater that integrates information across steps, and a cross-attention Injector that writes the updated memory back into the backbone. We train these modules with a dedicated K-step unrolling pipeline to learn multi-step dynamics. MetaState adds only {sim}0.6% trainable parameters while keeping the backbone frozen, and consistently improves reasoning performance over frozen baselines on mathematical reasoning and code generation benchmarks, with an average gain of 4.5% across all evaluations.

The ongoing evolution of language models has led to the development of large-scale architectures that demonstrate exceptional performance across a wide range of tasks. However, these models come with significant computational and energy demands, as well as potential privacy implications. In this context, Small Reasoning Language Models (SRLMs) with approximately 0.5 billion parameters present a compelling alternative due to their remarkable computational efficiency and cost effectiveness, particularly in resource-constrained environments. Despite these advantages, the limited capacity of 0.5 billion parameter models poses challenges in handling complex tasks such as mathematical reasoning and code generation. This research investigates various training strategies, including supervised fine-tuning (SFT), knowledge distillation (KD), and reinforcement learning (RL), as well as their hybrid implementations, to enhance the performance of 0.5B SRLMs. We analyze effective methodologies to bridge the performance gap between SRLMS and larger models and present insights into optimal training pipelines tailored for these smaller architectures. Through extensive experimental validation and analysis, our work aims to provide actionable recommendations for maximizing the reasoning capabilities of 0.5B models.

Recent years have seen considerable advancements in multi-step reasoning with Large Language Models (LLMs). The previous studies have elucidated the merits of integrating feedback or search mechanisms during model inference to improve the reasoning accuracy. The Process-Supervised Reward Model (PRM), typically furnishes LLMs with step-by-step feedback during the training phase, akin to Proximal Policy Optimization (PPO) or reject sampling. Our objective is to examine the efficacy of PRM in the inference phase to help discern the optimal solution paths for multi-step tasks such as mathematical reasoning and code generation. To this end, we propose a heuristic greedy search algorithm that employs the step-level feedback from PRM to optimize the reasoning pathways explored by LLMs. This tailored PRM demonstrated enhanced results compared to the Chain of Thought (CoT) on mathematical benchmarks like GSM8K and MATH. Additionally, to explore the versatility of our approach, we develop a novel method to automatically generate step-level reward dataset for coding tasks and observed similar improved performance in the code generation tasks. Thus highlighting the robust nature of our reward-model-based approach to inference for reasoning tasks.

Diffusion Language Models (DLMs) offer order-agnostic generation that can explore many possible decoding trajectories. However, current decoding methods commit to a single trajectory, limiting exploration in trajectory space. We introduce Order-Token Search to explore this space through jointly searching over generation order and token values. Its core is a likelihood estimator that scores denoising actions, enabling stable pruning and efficient exploration of diverse trajectories. Across mathematical reasoning and coding benchmarks, Order-Token Search consistently outperforms baselines on GSM8K, MATH500, Countdown, and HumanEval (3.1%, 3.8%, 7.9%, and 6.8% absolute over backbone), matching or surpassing diffu-GRPO post-trained d1-LLaDA. Our work establishes joint search as a key component for advancing decoding in DLMs.

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.

Reinforcement Learning with Verifiable Rewards (RLVR) has markedly enhanced the reasoning abilities of large language models (LLMs). Its success, however, largely depends on strong base models with rich world knowledge, yielding only modest improvements for small-size language models (SLMs). To address this limitation, we investigate Guided GRPO, which injects ground-truth reasoning steps into roll-out trajectories to compensate for SLMs' inherent weaknesses. Through a comprehensive study of various guidance configurations, we find that naively adding guidance delivers limited gains. These insights motivate G^2RPO-A, an adaptive algorithm that automatically adjusts guidance strength in response to the model's evolving training dynamics. Experiments on mathematical reasoning and code-generation benchmarks confirm that G^2RPO-A substantially outperforms vanilla GRPO. Our code and models are available at https://github.com/T-Lab-CUHKSZ/G2RPO-A.

Masked diffusion language models (MDLMs) offer the potential for parallel token generation, but most open-source MDLMs decode fewer than 5 tokens per model forward pass even with sophisticated sampling strategies. As a result, their sampling speeds are often comparable to AR + speculative decoding schemes, limiting their advantage over mainstream autoregressive approaches. Existing distillation-based accelerators (dParallel, d3LLM) finetune MDLMs on trajectories generated by a base model, which can become off-policy during finetuning and restrict performance to the quality of the base model's samples. We propose dUltra, an on-policy reinforcement learning framework based on Group Relative Policy Optimization (GRPO) that learns unmasking strategies for efficient parallel decoding. dUltra introduces an unmasking planner head that predicts per-token unmasking likelihoods under independent Bernoulli distributions. We jointly optimize the base diffusion LLM and the unmasking order planner using reward signals combining verifiable reward, distillation reward, and the number of unmasking steps. Across mathematical reasoning and code generation tasks, dUltra improves the accuracy--efficiency trade-off over state-of-the-art heuristic and distillation baselines, moving towards achieving ``diffusion supremacy'' over autoregressive models.

As post-training optimization becomes central to improving large language models, we observe a persistent saturation bottleneck: once models grow highly confident, further training yields diminishing returns. While existing methods continue to reinforce target predictions, we find that informative supervision signals remain latent in models' own historical weak states. Motivated by this observation, we propose WMSS (Weak Agents Can Make Strong Agents Stronger), a post-training paradigm that leverages weak checkpoints to guide continued optimization. By identifying recoverable learning gaps via entropy dynamics and reinforcing them through compensatory learning, WMSS enables strong agents to improve beyond conventional post-training saturation. Experiments on mathematical reasoning and code generation datasets show that agents trained with our approach achieve effective performance improvements, while incurring zero additional inference cost.

Model merging has emerged as a cost-effective approach for consolidating the capabilities of multiple LLMs without retraining. However, existing merging techniques, largely based on linear parameter arithmetic or optimization, struggle when applied to Mixture-of-Experts (MoE) architectures. We identify a critical failure mode in MoE merging, termed routing breakdown, in which the merged router fails to dispatch tokens to suitable experts. Routing breakdown stems from the sensitivity of the non-linear softmax and discrete Top-k routing mechanisms to parameter perturbations from merging, a sensitivity further amplified by load-balancing constraints imposed during MoE pretraining. Because fine-tuned experts exhibit distinct specializations, even modest misrouting can cause severe performance degradation. To address this issue, we propose Hessian-Aware Router Calibration (HARC), a training-free framework that leverages second-order curvature information to realign the merged router. This approach admits a closed-form solution that can be efficiently solved using a matrix-free conjugate gradient method. Experiments on mathematical reasoning and code generation tasks show that HARC effectively mitigates routing breakdown across diverse MoE merging baselines and leads to substantial performance improvements. Our code is available at https://github.com/huangcb01/HARC.

On-policy distillation (OPD) improves LLM reasoning by training a student model on its own generated outputs, but standard OPD treats all student-generated outputs (SGOs) equally regardless of their informativeness. We observe a consistent asymmetry in controlled filtering experiments: in both OPD and on-policy self distillation (OPSD), training only on incorrect SGOs outperforms training only on correct ones. Our further analysis suggests that models trained on correct-only SGOs tend to generate shorter reasoning traces and show weaker reflection behavior, while incorrect SGOs better preserve exploratory reasoning near the model's capability boundary. To exploit this signal without requiring full answer-containing rollouts, we introduce ReNIO, which Reweights Negative trajectory Importance for LLM On-policy distillation. By using the student-to-teacher probability ratio, ReNIO identifies pivotal tokens leading to wrong reasoning traces and aggregates their information into a normalized sample weight, inherently assigning larger weights to likely negative trajectories without observing the correctness of final-answer. Since Re-NIO only uses prefix-conditioned token probabilities, it preserves OPD's prefix training advantage over full-rollout reinforcement learning. Across both mathematical reasoning and code generation tasks, ReNIO improves both OPD and OPSD, with representative relative gains of up to 8.90% for Qwen3-1.7B and 10.00% for R1-Distill-Qwen-7B on mathematical reasoning benchmarks. Code repo: https://github.com/BDML-lab/ReNIO.

LLM agents increasingly rely on memory mechanisms to reuse knowledge from past problem-solving experiences. However, existing methods typically construct memory for a single agent and reuse it with the same underlying model, tightly coupling stored knowledge to model-specific reasoning styles. In heterogeneous deployments, where agents may be instantiated with backbone models of different sizes, architectures, or specializations, this raises a key question: can a single memory system be shared across agents with different backbone models? We find that naive cross-model memory transfer can degrade performance, because stored memories often entangle task-relevant knowledge with model-specific biases. To address this challenge, we propose MemCollab, a collaborative memory framework that builds shared cross-model memory by contrasting reasoning trajectories generated by different model-based agents on the same task. Through this contrastive process, MemCollab distills abstract reasoning constraints that capture shared task-level invariants while suppressing model-specific artifacts. We further introduce a task-aware retrieval mechanism that conditions memory access on task category, ensuring that only relevant constraints are retrieved at inference time. Experiments on mathematical reasoning and code generation benchmarks show that MemCollab consistently improves both accuracy and inference-time efficiency across diverse agents, including settings with different model families. These results demonstrate that collaboratively constructed cross-model memory can serve as a shared reasoning resource for heterogeneous LLM-based agents.

Large Language Models (LLMs) have demonstrated exceptional performance across diverse domains but are increasingly constrained by high inference latency. Early Exit has emerged as a promising solution to accelerate inference by dynamically bypassing redundant layers. However, in decoder-only architectures, the efficiency of Early Exit is severely bottlenecked by the KV Cache Absence problem, where skipped layers fail to provide the necessary historical states for subsequent tokens. Existing solutions, such as recomputation or masking, either introduce significant latency overhead or incur severe precision loss, failing to bridge the gap between theoretical layer reduction and practical wall-clock speedup. In this paper, we propose River-LLM, a training-free framework that enables seamless token-level Early Exit. River-LLM introduces a lightweight KV-Shared Exit River that allows the backbone's missing KV cache to be naturally generated and preserved during the exit process, eliminating the need for costly recovery operations. Furthermore, we utilize state transition similarity within decoder blocks to predict cumulative KV errors and guide precise exit decisions. Extensive experiments on mathematical reasoning and code generation tasks demonstrate that River-LLM achieves 1.71 to 2.16 times of practical speedup while maintaining high generation quality.

Optimizing LLM-based agentic workflows is challenging for scaling AI capabilities. Current methods rely on coarse, end-to-end evaluation signals and lack fine-grained signals on where to refine, often resulting in inefficient or low-impact modifications. To address these limitations, we propose JudgeFlow, an Evaluation-Judge-Optimization-Update pipeline. We incorporate reusable, configurable logic blocks into agentic workflows to capture fundamental forms of logic. On top of this abstraction, we design a dedicated Judge module that inspects execution traces particularly failed runs and assigns rank-based responsibility scores to problematic blocks. These fine-grained diagnostic signals are then leveraged by an LLM-based optimizer, which focuses modifications on the most problematic block in the workflow. Our approach improves sample efficiency, enhances interpretability through block-level diagnostics, and provides a scalable foundation for automating increasingly complex agentic workflows. We evaluate JudgeFlow on mathematical reasoning and code generation benchmarks, where JudgeFlow achieves superior performance and efficiency compared to existing methods.

Looped Transformers improve parameter efficiency by repeatedly applying shared Transformer blocks over multiple recurrent loops, increasing computational depth without increasing the parameter count. However, KV cache memory still scales with the number of loops, becoming a key memory bottleneck that limits batch size and inference throughput. KV cache quantization can alleviate this bottleneck, but existing methods often suffer substantial accuracy degradation at aggressive low-precision regimes. We observe that looped Transformers offer a unique opportunity: KV states across loops are highly similar. Based on this observation, we propose ResidualQuant, which uses the final-loop KV states as a reference and represents the remaining loops with low-precision residuals. Our method further combines least-square scaling and rotations applied to the residuals, as well as loop-wise mixed precision, to enable accurate quantization down to INT2 while retaining efficient reconstruction. Across multiple looped Transformer models and mathematical reasoning and code generation benchmarks, ResidualQuant consistently improves the accuracy-memory tradeoff over state-of-the-art rotation-based KV quantization. In particular, our method retains accuracy close to BF16 under mixed-precision settings while reducing theoretical KV storage by 80.7%, achieving up to 13.0% higher accuracy than the rotation-based baseline at the same memory budget. On an RTX 5090, the reduced KV memory traffic improves fixed-batch decode throughput by up to 2.73x, while the smaller memory footprint enables up to 2x larger batches, improving peak throughput by up to 4.15x.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as an effective paradigm for improving the reasoning capabilities of large language models. However, RLVR training is often hindered by sparse binary rewards and weak credit assignment, resulting in ambiguous optimization signals and underutilization of the useful information embedded in failed trajectories. To address this challenge, we propose Correction-Oriented Policy Optimization (CIPO), a simple and effective extension to RLVR that converts on-policy failed trajectories into correction-oriented supervision, without relying on any external signals. By jointly optimizing correction samples derived from the model's own failed attempts together with the standard RLVR objective, CIPO improves learning effectiveness while explicitly enhancing the model's ability to correct its own errors. Extensive experiments across 11 benchmarks spanning mathematical reasoning and code generation demonstrate that CIPO consistently and significantly outperforms strong baselines in both reasoning and correction performance. Moreover, CIPO yields stronger pass@K gains, indicating that it improves the model's intrinsic reasoning capacity rather than merely redistributing probability mass over existing correct answers.

On-Policy Distillation (OPD) trains a student on its own generations using teacher feedback. With different tokenizers, comparing teacher and student predictions requires alignment at both sequence and vocabulary levels. In this paper, we examine whether expanding this alignment coverage improves learning. Across three heterogeneous teacher--student pairs on mathematical reasoning and code generation, strict 1:1 groups already cover most student-generated tokens despite substantial vocabulary mismatch. On responses sampled from the students before distillation, the shared vocabulary retains nearly all teacher and student probability mass at strictly aligned positions on average. Restricting reverse KL to a student-selected top-16 subset of the shared vocabulary at each strict position achieves accuracy comparable to full shared-vocabulary OPD, outperforming the evaluated cross-tokenizer baselines. Adding mean squared error supervision on span log-probabilities in mismatch groups gives complete supervision coverage, yet reduces accuracy. At checkpoints from training with only the strict loss, the span gradients show weak or negative directional agreement with the strict gradients and grow in magnitude relative to them. These diagnostics may help explain the accuracy drop from adding span supervision. Our findings motivate a shift from maximizing alignment coverage to prioritizing supervision reliability: compact supervision at strict positions can be more effective than broader coverage that introduces weakly aligned or conflicting training signals.

Exploration has been a focus of reinforcement learning research for a long time. Recently, there has been growing evidence that it is also an important ingredient in LLM reinforcement learning recipes that can significantly impact downstream performance. Many existing methods control exploration in the action-space, for example, using temperature scaling. However, these methods cannot reorder tokens but only influence the variance in the output distribution. This limits exploration and can lead to divergence or stalled training. Here, we investigate parameter-space exploration, where rollouts are generated by sampling different policies from a posterior that may each explore different rollouts. Sampling less or more diverse policies is then a complementary control lever over exploration. We introduce a family of methods called Perturbed Parameter Policy Optimization (3PO) which use different sampling strategies and different rollout grouping for reward estimation. Experiments on OLMo-3-1025-7B and Qwen2.5-Math-7B across mathematical reasoning and code generation tasks show that these approaches consistently improve average downstream performance over standard GRPO at a near-identical FLOPs cost. Moreover, using multiple parameter samples consistently produces fewer zero-advantage groups and malformed or incorrect rollouts during training than GRPO and action-space baselines. Overall, our work presents evidence that parameter-space exploration can improve reinforcement learning for LLMs.

Reinforcement learning with verifiable rewards (RLVR) has substantially improved the reasoning capabilities of large language models on tasks such as mathematical reasoning and code generation. However, most RLVR methods assign a scalar outcome reward to an entire trajectory, resulting in sparse supervision and limited token-level credit assignment. On-policy distillation (OPD) provides denser supervision by distilling token-level distributions from a stronger teacher model, but requires an additional teacher and typically assumes a shared vocabulary. On-policy self-distillation (OPSD) removes this dependency by conditioning the same model on privileged information to construct a teacher policy. However, directly matching the teacher distribution may cause information leakage and unstable optimization. RLSD avoids direct matching by using the teacher signal only to modulate update magnitudes, but it cannot provide an explicit correction direction when the sampled reasoning fails. To address this tradeoff, we introduce H^{2}SD, a hybrid hindsight self distillation framework that uses the teacher differently according to trajectory correctness. For successful trajectories, the teacher receives the student response confirmed as correct together with a rephrasing instruction, and its probabilities on the original response tokens are used to modulate update magnitudes without changing the direction determined by the reward. For failed trajectories, we condition the teacher on a reference hint containing key reasoning steps and a verified answer, and minimize the reverse KL divergence from the student to the teacher. Experiments on multiple challenging reasoning benchmarks show that H^2SD consistently outperforms representative RLVR, OPSD, and RLSD baselines while maintaining stable optimization and favorable generation efficiency.

Group Relative Policy Optimization (GRPO) has proven effective in RLVR by using outcome-based rewards. While fine-grained dense rewards can theoretically improve performance, we reveal that under practical sampling budgets, Monte Carlo estimation yields high-variance and sign-inconsistent advantages for early reasoning tokens, paradoxically underperforming outcome-only GRPO. We propose Skip-Connected Optimization (SKPO), which decomposes reasoning into upstream and downstream phases: upstream receives dense rewards from downstream Monte Carlo sampling with single-stream optimization; downstream maintains group-relative optimization, where a skip connection concatenates the upstream segment with the original problem, enabling the model to leverage helpful upstream reasoning while preserving the freedom to bypass flawed reasoning through direct problem access. Experiments demonstrate improvements of 3.91% and 6.17% relative gains over the strongest baselines on Qwen2.5-Math-7B and Llama-3.2-3B respectively across mathematical benchmarks and out-of-domain tasks including general reasoning and code generation. Further analysis reveals an implicit advantage: SKPO generates trajectories with higher intermediate-step quality even when matched for final correctness.

Diffusion large language models (dLLMs) generate text by iteratively denoising a masked sequence, offering a parallel alternative to autoregressive models, but eliciting strong reasoning through post-training remains difficult: supervised fine-tuning is off-policy and suffers from exposure bias, while reinforcement learning gives only sparse, sequence-level rewards and is hard to apply without tractable sequence likelihoods. On-policy self-distillation (OPSD) offers a promising alternative, using one model as both student and teacher to provide dense, token-level, on-policy supervision, but its effectiveness hinges on giving the teacher privileged information (PI) - typically an instance-specific ground-truth reference unavailable at inference - so the student ends up distilling a weak PI-free consensus policy that yields little improvement on dLLM reasoning. We introduce dOPSD, which instead derives the teacher's privilege directly from the student's own denoising trajectory, evaluating masked positions using later, more-decoded steps of that same trajectory rather than an external label, so the teacher's advantage emerges from the model's own decoding process; on Dream and LLaDA, dOPSD improves both in-domain math reasoning and out-of-domain code generation, outperforming supervised and on-policy baselines.

Reinforcement learning with verifiable rewards (RLVR) broadcasts a single response-level reward to every token, while on-policy distillation (OPD) scores each token against a stronger teacher for a dense advantage but caps performance at teacher quality and discourages exploration beyond it. Their complementarity makes combining RLVR and OPD promising, but we find that fusing the two advantages with a fixed coefficient triggers entropy collapse from two miscalibrations: a magnitude mismatch, where token-level OPD advantages can spike far beyond the bounded RLVR advantage and erase its signal, and a temporal mismatch, where sustained full-strength OPD keeps pulling the student toward the teacher and limits exploration needed to surpass it. We propose SAF, a Stable Advantage Fusion framework that resolves both issues via a lightweight, four-stage pipeline applied only to the OPD advantage: a sparsify-then-compress mechanism for magnitude control paired with a warm-up-then-anneal mechanism for temporal control, with each stage independently switchable and adding negligible overhead. Instantiating RLVR with GRPO, we evaluate SAF across seven mathematical reasoning and code generation benchmarks with Qwen3-1.7B/4B/8B: SAF avoids entropy collapse and consistently outperforms fixed-coefficient GRPO+OPD fusion, improving the aggregate score by 0.51-2.70% across all six model-domain settings while achieving more stable training.

Large language models serve applications with diverse inference budgets, but training and maintaining separate models for each budget incurs substantial costs. Recent Looped Transformers enable flexible inference computation through parameter reuse, but maintaining strong performance across different inference budgets remains an open challenge. We introduce Accordion, a mixture-of-experts-based model that supports five inference budgets through fixed-prefix layer looping. Each budget follows a pre-defined execution path, while the minimum budget retains the unlooped model. Joint pretraining samples one budget per micro-batch and optimizes its next-token prediction loss without additional cross-budget consistency objectives. We further preserve these inference budgets through supervised fine-tuning and reinforcement learning, where parameter-group merging and budget-matched multi-teacher on-policy distillation integrate three domain experts into one model. After 3.6T tokens of pretraining, increasing inference depth improves Accordion-6B-A0.5B's five-category macro-average across 37 tasks by 3.01 percentage points, with the middle budget capturing 89% of this gain using half as many additional layer executions. The final model surpasses each domain expert on its corresponding task-and-budget average, while achieving strong instruction-following performance across all budgets and retaining depth-dependent gains in mathematical reasoning and code generation.

This work studies how to adaptively recompute key-value (KV) caches for diffusion large language models (DLMs) to maximize prediction accuracy while minimizing decoding latency. Prior methods' decoders recompute QKV for all tokens at every denoising step and layer, despite KV states changing little across most steps, especially in shallow layers, leading to substantial redundancy. We make three observations: (1) distant {bf MASK} tokens primarily act as a length-bias and can be cached block-wise beyond the active prediction window; (2) KV dynamics increase with depth, suggesting that selective refresh starting from deeper layers is sufficient; and (3) the most-attended token exhibits the smallest KV drift, providing a conservative lower bound on cache change for other tokens. Building on these, we propose {bf Elastic-Cache}, a training-free, architecture-agnostic strategy that jointly decides {when} to refresh (via an attention-aware drift test on the most-attended token) and {where} to refresh (via a depth-aware schedule that recomputes from a chosen layer onward while reusing shallow-layer caches and off-window MASK caches). Unlike fixed-period schemes, Elastic-Cache performs adaptive, layer-aware cache updates for diffusion LLMs, reducing redundant computation and accelerating decoding with negligible loss in generation quality. Experiments on LLaDA-Instruct, LLaDA-1.5, and LLaDA-V across mathematical reasoning and code generation tasks demonstrate consistent speedups: 8.7times on GSM8K (256 tokens), 45.1times on longer sequences, and 4.8times on HumanEval, while consistently maintaining higher accuracy than the baseline. Our method achieves significantly higher throughput (6.8times on GSM8K) than existing confidence-based approaches while preserving generation quality, enabling practical deployment of diffusion LLMs.

Large Language Models (LLMs) have made substantial strides in structured tasks through Reinforcement Learning (RL), demonstrating proficiency in mathematical reasoning and code generation. However, applying RL in broader domains like chatbots and content generation -- through the process known as Reinforcement Learning from Human Feedback (RLHF) -- presents unique challenges. Reward models in RLHF are critical, acting as proxies that evaluate the alignment of LLM outputs with human intent. Despite advancements, the development of reward models is hindered by challenges such as computational heavy training, costly evaluation, and therefore poor reproducibility. We advocate for using embedding-based input in reward model research as an accelerated solution to those challenges. By leveraging embeddings for reward modeling, we can enhance reproducibility, reduce computational demands on hardware, improve training stability, and significantly reduce training and evaluation costs, hence facilitating fair and efficient comparisons in this active research area. We then show a case study of reproducing existing reward model ensemble research using embedding-based reward models. We discussed future avenues for research, aiming to contribute to safer and more effective LLM deployments.

Reinforcement Learning with Verifiable Rewards (RLVR) has achieved remarkable success in improving autoregressive models, especially in domains requiring correctness like mathematical reasoning and code generation. However, directly applying such paradigms to Diffusion Large Language Models (dLLMs) is fundamentally hindered by the intractability of exact likelihood computation, which forces existing methods to rely on high-variance approximations. To bridge this gap, we propose Likelihood-Free Policy Optimization (LFPO), a native framework that maps the concept of vector field flow matching to the discrete token space. Specifically, LFPO formulates alignment as geometric velocity rectification, which directly optimizes denoising logits via contrastive updates. This design effectively bypasses the errors inherent in likelihood approximation, yielding the precise gradient estimation. Furthermore, LFPO enforce consistency by predicting final solutions from intermediate steps, effectively straightening the probability flow to enable high-quality generation with significantly fewer iterations. Extensive experiments demonstrate that LFPO not only outperforms state-of-the-art baselines on code and reasoning benchmarks but also accelerates inference by approximately 20% through reduced diffusion steps.

Post-training has become a crucial step for unlocking the capabilities of large language models, with reinforcement learning (RL) emerging as a critical paradigm. Recent RL-based post-training has increasingly split into two paradigms: reinforcement learning from human feedback (RLHF), which optimizes models using human preference signals in target domains, and reinforcement learning from verifiable rewards (RLVR), which operates in verifier-backed environments. The latter has dominated recent reasoning-oriented post-training because it delivers stronger gains and higher efficiency on domain-specific tasks (e.g., reasoning). However, although in-domain RL training achieves promising performance, it still requires a substantial amount of GPU compute, which remains a major barrier to broad adoption. In this work, we study the generalization ability of RLHF learned from scratch from a small set of interactions in open-ended environments, and investigate whether the conversational abilities it explicitly acquires can implicitly transfer to downstream tasks such as mathematical reasoning and code generation, namely GRLO. Specifically, on Qwen3-4B-Base backbone, GRLO improves the average performance across all domains from 24.1 to 63.1 with only 5K prompts and 22.7 GPU hours, requiring about 46times less data and 68times less compute than a strong in-domain RLVR baseline. The resulting model is even competitive with Qwen's released post-trained models which required a much larger training cost. Notably, a subsequent in-domain RLVR stage brings only selective gains, mainly on harder competition-math benchmarks. We hope GRLO offers a simple and efficient recipe for building broadly capable post-trained models. Our code and data will be available at: https://github.com/SJY8460/GRLO{https://github.com/SJY8460/GRLO}.

Pretrained Large Language Models (LLMs) are prone to generating fluent yet factually incorrect text-a phenomenon known as hallucinations, undermining their reliability and utility in downstream tasks. We hypothesize that a generated text span's factuality is correlated with its representational instability across the model's internal layers. Based on this, we propose the CoCoA (Confusion and Consistency Aware) decoder, a novel, training-free decoding algorithm that mitigates hallucinations at inference time by listening to these signals in the middle layers. We propose two metrics to quantify this instability in the middle layers and use it to penalize outputs that exhibit high internal confusion, thereby steering the model towards more internally consistent and factually grounded outputs. We further propose a self-information gated variant, CoCoA-SIG, that dynamically modulates this penalty to selectively target high-surprise, unstable generations. Extensive experiments on diverse tasks, including question-answering, summarization, mathematical reasoning and code generation, demonstrate that CoCoA significantly improves factual correctness across multiple model families (e.g., Llama-3, Qwen-2.5, Mistral). By leveraging model-intrinsic signals, CoCoA offers an effective and broadly applicable method for enhancing the trustworthiness of LLMs at inference time, without requiring any model retraining.

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.

Large language models (LLMs) excel at complex reasoning tasks but remain computationally expensive, limiting their practical deployment. To address this, recent works have focused on distilling reasoning capabilities into smaller language models (sLMs) using chain-of-thought (CoT) traces from teacher LLMs. However, this approach struggles in scenarios requiring rare factual knowledge or precise computation, where sLMs often hallucinate due to limited capability. In this work, we propose Agent Distillation, a framework for transferring not only reasoning capability but full task-solving behavior from LLM-based agents into sLMs with retrieval and code tools. We improve agent distillation along two complementary axes: (1) we introduce a prompting method called first-thought prefix to enhance the quality of teacher-generated trajectories; and (2) we propose a self-consistent action generation for improving test-time robustness of small agents. We evaluate our method on eight reasoning tasks across factual and mathematical domains, covering both in-domain and out-of-domain generalization. Our results show that sLMs as small as 0.5B, 1.5B, 3B parameters can achieve performance competitive with next-tier larger 1.5B, 3B, 7B models fine-tuned using CoT distillation, demonstrating the potential of agent distillation for building practical, tool-using small agents. Our code is available at https://github.com/Nardien/agent-distillation.

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.

Large reasoning models achieve strong performance on challenging tasks, but their long chain-of-thought often incurs substantial inference costs by continuing to deliberate after sufficient information has been obtained. Existing reasoning compression methods primarily operate on reasoning trajectories, paying less attention to the local decision of whether further reasoning is necessary given the current state. We propose LoCo, a local preference learning method for reasoning compression. LoCo detects candidate overthinking states from the model's own traces, freezes the corresponding reasoning prefixes, regenerates shorter verified continuations from the same states, and trains on the resulting state-conditioned preference pairs with DPO. This formulation teaches the model when to continue reasoning and when to conclude, rather than simply encouraging shorter trajectories overall. We construct training data entirely from self-generated mathematical traces and evaluate LoCo on five mathematical and three code generation benchmarks. LoCo reduces completion length by 41.7\%--45.4\% on mathematics and 23.4\%--36.3\% on code, while maintaining or improving accuracy and achieving the best AUC across all code benchmarks. LoCo consistently improves the accuracy-efficiency trade-off on mathematics and transfers effectively to code generation despite math-only training, demonstrating the cross-domain generality of local preference learning for reasoning compression. The code and model will be publicly released at https://github.com/MuyaoYuan/LoCo.

The recent paradigm shift towards training large language models (LLMs) using DeepSeek-R1-Zero-style reinforcement learning (RL) on verifiable rewards has led to impressive advancements in code and mathematical reasoning. However, this methodology is limited to tasks where rule-based answer verification is possible and does not naturally extend to real-world domains such as chemistry, healthcare, engineering, law, biology, business, and economics. Current practical workarounds use an additional LLM as a model-based verifier; however, this introduces issues such as reliance on a strong verifier LLM, susceptibility to reward hacking, and the practical burden of maintaining the verifier model in memory during training. To address this and extend DeepSeek-R1-Zero-style training to general reasoning domains, we propose a verifier-free method (VeriFree) that bypasses answer verification and instead uses RL to directly maximize the probability of generating the reference answer. We compare VeriFree with verifier-based methods and demonstrate that, in addition to its significant practical benefits and reduced compute requirements, VeriFree matches and even surpasses verifier-based methods on extensive evaluations across MMLU-Pro, GPQA, SuperGPQA, and math-related benchmarks. Moreover, we provide insights into this method from multiple perspectives: as an elegant integration of training both the policy and implicit verifier in a unified model, and as a variational optimization approach. Code is available at https://github.com/sail-sg/VeriFree.

While large language models (LLMs) are increasingly deployed as dense retrievers, the impact of their domain-specific specialization on retrieval effectiveness remains underexplored. This investigation systematically examines how task-specific adaptations in LLMs influence their retrieval capabilities, an essential step toward developing unified retrievers capable of handling text, code, images, and multimodal content. We conduct extensive experiments with eight Qwen2.5 7B LLMs, including base, instruction-tuned, code/math-specialized, long reasoning, and vision-language models across zero-shot retrieval settings and the supervised setting. For the zero-shot retrieval settings, we consider text retrieval from the BEIR benchmark and code retrieval from the CoIR benchmark. Further, to evaluate supervised performance, all LLMs are fine-tuned on the MS MARCO dataset. We find that mathematical specialization and the long reasoning capability cause consistent degradation in three settings, indicating conflicts between mathematical reasoning and semantic matching. The vision-language model and code-specialized LLMs demonstrate superior zero-shot performance compared to other LLMs, even surpassing BM25 on the code retrieval task, and maintain comparable performance to base LLMs in supervised settings. These findings suggest promising directions for the unified retrieval task leveraging cross-domain and cross-modal fusion.

The emergence of large reasoning models (LRMs) has transformed Natural Language Processing by excelling in complex tasks such as mathematical problem-solving and code generation. These models leverage chain-of-thought (CoT) processes, enabling them to emulate human-like reasoning strategies. However, the advancement of LRMs is hindered by the lack of comprehensive CoT datasets. Current resources often fail to provide extensive reasoning problems with coherent CoT processes distilled from multiple teacher models and do not account for multifaceted properties describing the internal characteristics of CoTs. To address these challenges, we introduce OmniThought, a large-scale dataset featuring 2 million CoT processes generated and validated by two powerful LRMs as teacher models. Each CoT process in OmniThought is annotated with novel Reasoning Verbosity (RV) and Cognitive Difficulty (CD) scores, which describe the appropriateness of CoT verbosity and cognitive difficulty level for models to comprehend these reasoning processes. We further establish a self-reliant pipeline to curate this dataset. Extensive experiments using Qwen2.5 models of various sizes demonstrate the positive impact of our proposed scores on LRM training effectiveness. Based on the proposed OmniThought dataset, we further train and release a series of high-performing LRMs, specifically equipped with stronger reasoning abilities and optimal CoT output length and difficulty level. Our contributions significantly enhance the development and training of LRMs for solving complex tasks.

Improving the reasoning abilities of large language models (LLMs) has largely relied on iterative self-training with model-generated data. While effective at boosting accuracy, existing approaches primarily reinforce successful reasoning paths, incurring a substantial calibration cost: models become overconfident and lose the ability to represent uncertainty. This failure has been characterized as a form of model collapse in alignment, where predictive distributions degenerate toward low-variance point estimates. We address this issue by reframing reasoning training as an epistemic learning problem, in which models must learn not only how to reason, but also when their reasoning should be trusted. We propose epistemically-calibrated reasoning (EpiCaR) as a training objective that jointly optimizes reasoning performance and calibration, and instantiate it within an iterative supervised fine-tuning framework using explicit self-evaluation signals. Experiments on Llama-3 and Qwen-3 families demonstrate that our approach achieves Pareto-superiority over standard baselines in both accuracy and calibration, particularly in models with sufficient reasoning capacity (e.g., 3B+). This framework generalizes effectively to OOD mathematical reasoning (GSM8K) and code generation (MBPP). Ultimately, our approach enables a 3X reduction in inference compute, matching the K=30 performance of STaR with only K=10 samples in capable models.

Continuous diffusion generates complete reasoning solutions through iterative refinement in latent space. We introduce Latent Flow Reasoning Models (LFRMs), an ELF-based training and inference recipe. Our experiments show that accurate decoding alone does not ensure strong reasoning performance. We therefore learn compact representations from multiple layers of a strong autoregressive teacher. Their decomposition also enables asynchronous denoising at different rates. We show that prompt encodings need only preserve the information required for the correct text-conditional score, rather than exactly match teacher features, and use a staged curriculum to learn a compact prompt encoder that replaces the teacher Transformer at inference. We adapt DiffusionNFT to learned self-conditioning guidance and incorporate gold-solution endpoints to supplement sparse rewards. Our supervised models outperform reported results from recent continuous-diffusion baselines at comparable backbone scales on mathematical reasoning and HumanEval code generation. With a 638M-parameter denoising backbone and learned prompt conditioning, post-NFT LFRM-L achieves 63.74% pass@1 on GSM8K and 24.6% on MATH500 at 64 denoising steps, and 32.85% on HumanEval and 30.18% on HumanEval+ at 128 denoising steps. Code will be available at: https://github.com/chengxiang/LFRM

Recent advances in large language models (LLMs), particularly those enhanced through reinforced post-training, have demonstrated impressive reasoning capabilities, as exemplified by models such as OpenAI o1 and DeepSeek-R1. However, these capabilities are predominantly benchmarked on domains like mathematical problem solving and code generation -- leaving open the question of whether such reasoning skills generalize to complex, real-world scenarios. In this paper, we introduce LocationReasoner, a benchmark designed to evaluate LLMs' reasoning abilities in the context of real-world site selection, where models must identify feasible locations by reasoning over diverse and complicated spatial, environmental, and logistical constraints. The benchmark comprises over 300 carefully crafted queries of varying difficulty levels, supported by a sandbox environment with in-house tools for constraint-based location search. Extensive evaluations reveal that state-of-the-art reasoning models offer limited improvement over their non-reasoning predecessors in real-world contexts, with even the latest OpenAI o4 model failing on 30% of site selection tasks. Moreover, agentic strategies such as ReAct and Reflexion often suffer from over-reasoning, leading to worse outcomes than direct code-generation prompting. With key limitations of LLMs in holistic and non-linear reasoning highlighted, we release LocationReasoner to foster the development of LLMs and agents capable of robust, grounded reasoning in real-world decision-making tasks. Codes and data for our benchmark are available at https://github.com/miho-koda/LocationReasoner.

Looped Language Models (LoopLMs) offer a parameter efficient approach to scaling reasoning by reusing shared parameters across recurrent computation steps. Despite their promise, effective post-training of LoopLMs remains challenging. Existing approaches either provide reward based supervision that is sparse or costly to extend across loops, or rely on external teachers or privileged information, leading to limited teacher availability or teacher-student context mismatch. To address these limitations, we introduce LoopOPD, a cross-loop on-policy distillation framework that uses additional recurrent computation within a LoopLM as its own source of supervision. LoopOPD uses a frozen terminal loop policy as a compute privileged teacher for an intermediate loop student on student generated rollouts, providing dense supervision without an external teacher or privileged information. We further propose Dynamic LoopOPD (D-LoopOPD), which continually refreshes the terminal loop teacher as the shared model parameters are updated, enabling recurrent self-improvement. We characterize how distillation updates propagate across loop depths and derive sufficient conditions under which a single update yields simultaneous local improvement at both loop depths. Experiments on Ouro-Thinking models show that LoopOPD improves mathematical reasoning, while D-LoopOPD yields further gains through dynamic teacher updates. Despite being trained only on mathematical data, the resulting models also improve on general reasoning and code generation benchmarks, demonstrating that recurrent computation can serve as an effective source of supervision for LoopLMs. Our code and model checkpoints will be released upon acceptance.

Diffusion language models (DLMs) have recently emerged as a strong alternative to autoregressive models by enabling parallel text generation. To improve inference efficiency and KV-cache compatibility, prior work commonly adopts block-based diffusion, decoding tokens block by block. However, this paradigm suffers from a structural limitation that we term Boundary-Induced Context Truncation (BICT): undecoded tokens near block boundaries are forced to commit without access to nearby future context, even when such context could substantially reduce uncertainty. This limitation degrades decoding certainty and generation quality, especially for tasks requiring precise reasoning, such as mathematical problem solving and code generation. We propose Deferred Commitment Decoding (DCD), a novel, training-free decoding strategy that mitigates this issue. DCD maintains a certainty-aware sliding window over masked tokens, resolving low-uncertainty tokens early while deferring high-uncertainty tokens until sufficient contextual evidence becomes available. Extensive experiments across multiple diffusion language models, benchmarks, and caching configurations show that DCD improves generation accuracy by 1.73% with comparable time on average compared to fixed block-based diffusion methods, with the most significant improvement reaching 16.5%. These results demonstrate that deferring token commitment based on uncertainty is a simple yet effective principle for improving both the quality and efficiency of diffusion language model decoding.

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.

Recent advancements in language models have led to significant improvements in mathematical reasoning across various benchmarks. However, most of these benchmarks rely on automatic evaluation methods that only compare final answers using heuristics, without verifying the underlying reasoning steps. This limitation results in false positive solutions, where models may produce correct final answers but with flawed deduction paths. In this paper, we systematically examine the prevalence of false positive solutions in mathematical problem solving for language models. We analyze the characteristics and extent of this issue across different open-source models, datasets of varying difficulty levels, and decoding strategies. Specifically, we explore how false positives influence the inference time scaling behavior of language models. Our experimental results reveal that: (1) false positive solutions persist across different models, datasets, and decoding methods, (2) sampling-based inference time scaling methods do not alleviate the problem, and (3) the pass@N evaluation metric is more susceptible to false positives, suggesting a significantly lower scaling ceiling than what automatic evaluations indicate. Additionally, we analyze specific instances of false positives and discuss potential limitations in self-improvement techniques and synthetic data generation under such conditions. Our data and code are publicly available at https://github.com/Wloner0809/False-Positives-in-Math.

This paper presents MathBode, a dynamic diagnostic for mathematical reasoning in large language models (LLMs). Instead of one-shot accuracy, MathBode treats each parametric problem as a system: we drive a single parameter sinusoidally and fit first-harmonic responses of model outputs and exact solutions. This yields interpretable, frequency-resolved metrics -- gain (amplitude tracking) and phase (lag) -- that form Bode-style fingerprints. Across five closed-form families (linear solve, ratio/saturation, compound interest, 2x2 linear systems, similar triangles), the diagnostic surfaces systematic low-pass behavior and growing phase lag that accuracy alone obscures. We compare several models against a symbolic baseline that calibrates the instrument (G approx 1, phi approx 0). Results separate frontier from mid-tier models on dynamics, providing a compact, reproducible protocol that complements standard benchmarks with actionable measurements of reasoning fidelity and consistency. We open-source the dataset and code to enable further research and adoption.

Recent large language models (LLMs) have shown indications of mathematical reasoning ability. However it has not been clear how they would fare on more challenging competition-level problems. And while self-generated verbalizations of intermediate reasoning steps (i.e., chain-of-thought prompting) have been shown to be helpful, whether LLMs can make use of helpful side information such as problem-specific hints has not been investigated before. In this paper, we propose a challenging benchmark dataset for enabling such analyses. The Concept and Hint-Annotated Math Problems (CHAMP) consists of high school math competition problems, annotated with concepts, or general math facts, and hints, or problem-specific tricks. These annotations allow us to explore the effects of additional information, such as relevant hints, misleading concepts, or related problems. This benchmark is difficult, with the best model only scoring 58.1% in standard settings. With concepts and hints, performance sometimes improves, indicating that some models can make use of such side information. We further annotate model-generated solutions for their correctness. Using this corpus, we find that models often arrive at the correct final answer through wrong reasoning steps. In addition, we test whether models are able to verify these solutions, and find that most models struggle. The dataset and code are available on the project website.

Mathematical reasoning is a hallmark of human intelligence, requiring logical deduction, symbolic manipulation, and abstract thinking. Recent multimodal large language models (MLLMs) have demonstrated strong performance on geometry problems through multi-step reasoning. To better emulate human problem-solving, intermediate steps can incorporate auxiliary visual constructions, such as additional lines or points, which improve geometric interpretation and educational clarity. In this work, we introduce the GeoMathCode, where programmatic representations serve as intermediate visual outputs. We further conduct an in-depth analysis of the underlying reasoning geometry. Experimental results show that reasoning and code generation steps can be disentangled in the latent space, while supervised fine-tuning (SFT) makes the reasoning manifold more structured and informative. Moreover, hierarchical syntactic code structures emerge as disentangled latent subspaces, and contain more mathematical symbolic information than visual representations.

Recent advancements in Chain-of-Thoughts (CoT) and Program-of-Thoughts (PoT) methods have greatly enhanced language models' mathematical reasoning capabilities, facilitating their integration into instruction tuning datasets with LLMs. However, existing methods for large-scale dataset creation require substantial seed data and high computational costs for data synthesis, posing significant challenges for scalability. We introduce InfinityMATH, a scalable instruction tuning dataset for programmatic mathematical reasoning. The construction pipeline emphasizes decoupling numbers from mathematical problems to synthesize number-independent programs, enabling efficient and flexible scaling while minimizing dependency on specific numerical values. Fine-tuning experiments with open-source language and code models, such as Llama2 and CodeLlama, demonstrate the practical benefits of InfinityMATH. These fine-tuned models, showed significant relative improvements on both in-domain and out-of-domain benchmarks, ranging from 184.7% to 514.3% on average. Additionally, these models exhibited high robustness on the GSM8K+ and MATH+ benchmarks, which are enhanced version of test sets with simply the number variations. InfinityMATH ensures that models are more versatile and effective across a broader range of mathematical problems. The data is available at https://huggingface.co/datasets/flagopen/InfinityMATH.

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.

Although large language models demonstrate emergent abilities in solving math word problems, there is a challenging task in complex multi-step mathematical reasoning tasks. To improve model performance on mathematical reasoning tasks, previous work has conducted supervised fine-tuning on open-source models by improving the quality and quantity of data. In this paper, we propose a novel approach, named Brain, to imitate human thought processes to enhance mathematical reasoning abilities, using the Frontal Lobe Model to generate plans, and then employing the Parietal Lobe Model to generate code and execute to obtain answers. First, we achieve SOTA performance in comparison with Code LLaMA 7B based models through this method. Secondly, we find that plans can be explicitly extracted from natural language, code, or formal language. Our code and data are publicly available at https://github.com/cyzhh/Brain.

Large Language Models (LLMs) have revolutionized both general natural language processing and domain-specific applications such as code synthesis, legal reasoning, and finance. However, while prior studies have explored individual model capabilities, a systematic cross-domain comparison that unifies linguistic, reasoning, and code understanding abilities remains underexplored. In this work, we present a comprehensive evaluation of five general-purpose and three code-specific state-of-the-art LLMs across six diverse benchmarks encompassing linguistic competence, mathematical reasoning, and trustworthiness. Additionally, we analyze model behavior on the CoNaLa dataset for code explanation, comparing natural language and code-specialized LLMs. Our findings reveal that models optimized for code (e.g., CodeLLaMA variants) exhibit strong reasoning and syntactic precision, that even for non-coding tasks can show measurable performance gains, in contrast to general-purpose models like Mistral-7B and Llama-3-8B.

Neurosymbolic approaches leveraging Large Language Models (LLMs) with formal methods have recently achieved strong results on mathematics-oriented theorem-proving benchmarks. However, success on competition-style mathematics does not by itself demonstrate the ability to construct proofs about real-world implementations. We address this gap with a benchmark derived from an industrial cryptographic library whose assembly routines are already verified in HOL Light. s2n-bignum is a library used at AWS for providing fast assembly routines for cryptography, and its correctness is established by formal verification. The task of formally verifying this library has been a significant achievement for the Automated Reasoning Group. It involved two tasks: (1) precisely specifying the correct behavior of a program as a mathematical proposition, and (2) proving that the proposition is correct. In the case of s2n-bignum, both tasks were carried out by human experts. In s2n-bignum-bench, we provide the formal specification and ask the LLM to generate a proof script that is accepted by HOL Light within a fixed proof-check timeout. To our knowledge, s2n-bignum-bench is the first public benchmark focused on machine-checkable proof synthesis for industrial low-level cryptographic assembly routines in HOL Light. This benchmark provides a challenging and practically relevant testbed for evaluating LLM-based theorem proving beyond competition mathematics. The code to set up and use the benchmark is available here: https://github.com/kings-crown/s2n-bignum-bench{s2n-bignum-bench}.

Multi-agent systems built from prompted large language models can improve multi-round reasoning, yet most existing pipelines rely on fixed, trajectory-wide communication patterns that are poorly matched to the stage-dependent needs of iterative problem solving. We introduce DyTopo, a manager-guided multi-agent framework that reconstructs a sparse directed communication graph at each round. Conditioned on the manager's round goal, each agent outputs lightweight natural-language query (need) and \key (offer) descriptors; DyTopo embeds these descriptors and performs semantic matching, routing private messages only along the induced edges. Across code generation and mathematical reasoning benchmarks and four LLM backbones, DyTopo consistently outperforms over the strongest baseline (avg. +6.2). Beyond accuracy, DyTopo yields an interpretable coordination trace via the evolving graphs, enabling qualitative inspection of how communication pathways reconfigure across rounds.

Games offer a compelling paradigm for developing general reasoning capabilities in language models, as they naturally demand strategic planning, probabilistic inference, and adaptive decision-making. However, existing self-play approaches rely solely on terminal game outcomes, providing no mechanism to distinguish transferable reasoning patterns from game-specific heuristics. We present STRATAGEM, which addresses two fundamental barriers to reasoning transfer: domain specificity, where learned patterns remain anchored in game semantics, and contextual stasis, where static game contexts fail to cultivate progressive reasoning. STRATAGEM selectively reinforces trajectories exhibiting abstract, domain-agnostic reasoning through a Reasoning Transferability Coefficient, while incentivizing adaptive reasoning development via a Reasoning Evolution Reward. Experiments across mathematical reasoning, general reasoning, and code generation benchmarks demonstrate substantial improvements, with particularly strong gains on competition-level mathematics where multi-step reasoning is critical. Ablation studies and human evaluation confirm that both components contribute to transferable reasoning.

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.

Test-time scaling for complex reasoning tasks shows that leveraging inference-time compute, by methods such as independently sampling and aggregating multiple solutions, results in significantly better task outcomes. However, a critical bottleneck is verification: sampling is only effective if correct solutions can be reliably identified among candidates. While existing approaches typically evaluate candidates independently via scalar scoring, we demonstrate that models are substantially stronger at pairwise self-verification. Leveraging this insight, we introduce V_1, a framework that unifies generation and verification through efficient pairwise ranking. V_1 comprises two components: V_1-Infer, an uncertainty-guided algorithm using a tournament-based ranking that dynamically allocates self-verification compute to candidate pairs whose relative correctness is most uncertain; and V_1-PairRL, an RL framework that jointly trains a single model as both generator and pairwise self-verifier, ensuring the verifier adapts to the generator's evolving distribution. On code generation (LiveCodeBench, CodeContests, SWE-Bench) and math reasoning (AIME, HMMT) benchmarks, V_1-Infer improves Pass@1 by up to 10% over pointwise verification and outperforms recent test-time scaling methods while being significantly more efficient. Furthermore, V_1-PairRL achieves 7--9% test-time scaling gains over standard RL and pointwise joint training, and improves base Pass@1 by up to 8.7% over standard RL in a code-generation setting.

Despite the rapid progress in the capabilities of Large Language Models (LLMs), they continue to have difficulty following relatively simple, unambiguous instructions, especially when compositions are involved. In this paper, we take inspiration from recent work that suggests that models may follow instructions better when they are expressed in pseudo-code. However, writing pseudo-code programs can be tedious and using few-shot demonstrations to craft code representations for use in inference can be unnatural for non-expert users of LLMs. To overcome these limitations, we propose fine-tuning LLMs with instruction-tuning data that additionally includes instructions re-expressed in pseudo-code along with the final response. We evaluate models trained using our method on 11 publicly available benchmarks comprising of tasks related to instruction-following, mathematics, and common-sense reasoning. We conduct rigorous experiments with 5 different models and find that not only do models follow instructions better when trained with pseudo-code, they also retain their capabilities on the other tasks related to mathematical and common sense reasoning. Specifically, we observe a relative gain of 3--19% on instruction-following benchmark, and an average gain of upto 14% across all tasks.

Multi-modality promises to unlock further uses for large language models. Recently, the state-of-the-art language model GPT-4 was enhanced with vision capabilities. We carry out a prompting evaluation of GPT-4V and five other baselines on structured reasoning tasks, such as mathematical reasoning, visual data analysis, and code generation. We show that visual Chain-of-Thought, an extension of Chain-of-Thought to multi-modal LLMs, yields significant improvements over the vanilla model. We also present a categorized analysis of scenarios where these models perform well and where they struggle, highlighting challenges associated with coherent multimodal reasoning.

This paper explores how theory can guide and enhance practical algorithms, using Low-Rank Adaptation (LoRA, Hu et al. 2022) in large language models as a case study. We rigorously prove that, under gradient descent, LoRA adapters align with specific singular subspaces of the one-step full fine-tuning gradient. This result suggests that, by properly initializing the adapters using the one-step full gradient, subspace alignment can be achieved immediately and applicable to both linear and nonlinear models. Building on our theory, we propose a theory-driven algorithm, LoRA-One, where the linear convergence (as well as generalization) is built and incorporating preconditioners theoretically helps mitigate the effects of ill-conditioning. Besides, our theory reveals connections between LoRA-One and other gradient-alignment-based methods, helping to clarify misconceptions in the design of such algorithms. LoRA-One achieves significant empirical improvements over LoRA and its variants across benchmarks in natural language understanding, mathematical reasoning, and code generation. Code is available at: https://github.com/YuanheZ/LoRA-One.

We introduce the latest series of TeleChat models: TeleChat2, TeleChat2.5, and T1, offering a significant upgrade over their predecessor, TeleChat. Despite minimal changes to the model architecture, the new series achieves substantial performance gains through enhanced training strategies in both pre-training and post-training stages. The series begins with TeleChat2, which undergoes pretraining on 10 trillion high-quality and diverse tokens. This is followed by Supervised Fine-Tuning (SFT) and Direct Preference Optimization (DPO) to further enhance its capabilities. TeleChat2.5 and T1 expand the pipeline by incorporating a continual pretraining phase with domain-specific datasets, combined with reinforcement learning (RL) to improve performance in code generation and mathematical reasoning tasks. The T1 variant is designed for complex reasoning, supporting long Chain-of-Thought (CoT) reasoning and demonstrating substantial improvements in mathematics and coding. In contrast, TeleChat2.5 prioritizes speed, delivering rapid inference. Both flagship models of T1 and TeleChat2.5 are dense Transformer-based architectures with 115B parameters, showcasing significant advancements in reasoning and general task performance compared to the original TeleChat. Notably, T1-115B outperform proprietary models such as OpenAI's o1-mini and GPT-4o. We publicly release TeleChat2, TeleChat2.5 and T1, including post-trained versions with 35B and 115B parameters, to empower developers and researchers with state-of-the-art language models tailored for diverse applications.

Self-evolving reasoning models learn from their own generated questions, yet repeated self-training can lead to performance collapse. In this paper, we investigate why performance deteriorates over successive rounds and how to sustain self-evolution. Our analysis identifies two recurring quality problems in self-generated questions: invalid questions and repeated variants of the same mathematical questions. First, invalid questions become more prevalent across rounds, and answer-consistency filtering further increases their proportion in training data. Second, existing question diversity controls based on lexical similarity can miss mathematically equivalent questions expressed in different ways, which leads to question diversity collapse in later training rounds. Building on these findings, we introduce R-Quest, which uses question validity and novelty feedback to guide self-evolution. We first train the solver to recognize and reject invalid questions, then use its judgments to guide questioner rewards and filter solver training data. To avoid question repetition, we use a frozen base model to compare sampled question pairs and provide novelty feedback. Empirically, our method consistently achieves the highest average performance on 12 benchmarks in mathematical reasoning, general-domain reasoning, and code generation across two model families. Additionally, R-Quest maintains stable performance gains over ten rounds of self-evolution, peaking in the final round and outperforming R-Zero by 17.32 points.

Low-Rank Adaptation (LoRA) is a widely used parameter-efficient fine-tuning method for foundation models, but it suffers from parameter interference, resulting in suboptimal performance. Although Mixture-of-Experts (MoE)-based LoRA variants show promise in mitigating intra-task correlations in single-task instruction tuning, they introduce additional router parameters and remain ineffective in multi-task model merging where inter-task interference arises. Inspired by the fly olfactory circuit, we propose FlyLoRA, an implicit MoE-based LoRA variant that introduces: (1) rank-wise expert activation in the up-projection matrix, and (2) an implicit router that unifies expert routing and down-projection, where a frozen sparse random projection matrix replaces the traditional dense trainable version. This design resolves the trade-off between intra-task decorrelation and computational efficiency by eliminating the need for an explicit router, while inherently mitigating inter-task interference due to the orthogonality property of random matrices. Extensive experiments across four domains -- general knowledge understanding, scientific question answering, mathematical reasoning, and code generation -- demonstrate consistent performance improvements over existing methods. Beyond empirical gains, FlyLoRA highlights how biological structures can inspire innovations in AI technologies. Code is available at https://github.com/gfyddha/FlyLoRA.

Agent systems based on large language models (LLMs) have shown great potential in complex reasoning tasks, but building efficient and generalizable workflows remains a major challenge. Most existing approaches rely on manually designed processes, which limits their adaptability across different tasks. While a few methods attempt automated workflow generation, they are often tied to specific datasets or query types and make limited use of intermediate feedback, reducing system robustness and reasoning depth. Moreover, their operations are typically predefined and inflexible. To address these limitations, we propose DyFlow, a dynamic workflow generation framework that adaptively constructs and adjusts reasoning procedures based on task requirements and real-time intermediate feedback, thereby enhancing cross-task generalization. DyFlow consists of two core components: a designer and an executor. The designer decomposes complex problems into a sequence of sub-goals defined by high-level objectives and dynamically plans the next steps based on intermediate outputs and feedback. These plans are then carried out by the executor, which executes each operation using dynamic operators with context-aware parameterization, enabling flexible and semantically grounded reasoning. We systematically evaluate DyFlow across diverse domains, including social reasoning, biomedical tasks, mathematical problem solving, and code generation. Results demonstrate that DyFlow significantly outperforms existing baselines, achieving substantial Pass@k improvements and exhibiting robust generalization across diverse domains. The code is publicly available at https://github.com/wyf23187/DyFlow.

Recent reinforcement learning (RL) methods improve LLM reasoning by optimizing discrete Chain-of-Thought (CoT) generation; however, exploration in token space often suffers from diversity collapse as policy entropy decreases due to mode elicitation behavior in discrete RL. To mitigate this issue, we propose Latent Diffusion Reasoning with Reinforcement Learning (LaDi-RL), a framework that conducts exploration directly in a continuous latent space, where latent variables encode semantic-level reasoning trajectories. By modeling exploration via guided diffusion, multi-step denoising distributes stochasticity and preserves multiple coexisting solution modes without mutual suppression. Furthermore, by decoupling latent-space exploration from text-space generation, we show that latent diffusion-based optimization is more effective than text-space policy optimization alone, while a complementary text policy provides additional gains when combined with latent exploration. Experiments on code generation and mathematical reasoning benchmarks demonstrate consistent improvements in both pass@1 and pass@k over discrete RL baselines, with absolute pass@1 gains of +9.4% on code generation and +5.7% on mathematical reasoning, highlighting diffusion-based latent RL as a principled alternative to discrete token-level RL for reasoning.

Masked diffusion language models predict tokens from a partially observed response canvas, enabling bidirectional conditioning and parallel token refinement. Yet standard masked-diffusion decoders use a rigid inference interface: the number of masked positions allocated to the answer is fixed before generation begins. Choosing this length is difficult. A short canvas can truncate reasoning or code, while a long canvas wastes computation and can perturb denoising. We introduce CARVE (Counterfactual-Aware Reveal with Verified Expansion), a training-free variable-length algorithm for masked diffusion LMs. Starting from a shorter canvas, CARVE can grow the response during decoding by inserting additional [MASK] positions. Rather than keeping every insertion, CARVE tests a candidate expanded canvas and asks a counterfactual question: would the model make similar predictions for the unresolved positions in the original canvas if the extra masked space were present? The inserted masks are kept only when they induce low Jensen-Shannon (JS) divergence on aligned unresolved positions. This makes length growth a verified stability decision rather than a pure confidence heuristic. CARVE applies without retraining to both full-canvas and blockwise diffusion decoders. Across code generation and mathematical reasoning benchmarks, CARVE consistently improves average performance over fixed-length baselines across all evaluated model families. Crucially, CARVE achieves these accuracy gains while reducing inference cost, reaching half the FLOPs of fixed-length decoding in some settings.

On-policy distillation (OPD) enables effective capability transfer between language models, yet the mechanisms underlying its failures are not fully understood. Across code generation and mathematical reasoning, OPD with larger-scale teachers exhibits early loss plateaus, with an average final loss reduction of 25.1% after 200 updates, compared with 96.2% for self-RL teachers, obtained by further reinforcement learning (RL) training of the initial student. To understand this difference, we analyze OPD as an idealized continuous-time dynamical system in the small-learning-rate limit. Our training-log diagnostics associate these plateaus with an early decline in a gradient-based learning-signal proxy while substantial loss remains; these measurements do not establish why the underlying gradient weakens. We further prove a local recovery guarantee for teachers sufficiently close to the initial student in a shared parameterization under regularity conditions, offering a conditional explanation for the success of self-RL teachers in our experiments. Across runs with and without loss plateaus, we observe small relative parameter changes (0.025-0.098%) and high similarity between the student's representations before and after OPD (linear CKA >0.98 across layers). These observations suggest that limited representation adaptation may contribute to learning-signal collapse, a hypothesis that remains to be tested. Code is available at https://github.com/leizhao7/opd-learning-signals.

LLMs are widely used for code generation and mathematical reasoning tasks where they are required to generate structured output. They either need to reason about code, generate code for a given specification, or reason using programs of thought. The typical approach to code generation is to prompt the model and generate samples until an appropriate program is obtained. Within this process, sampling n programs from the language model requires n GPU compute-intensive generations which becomes prohibitively expensive for larger values of n. In this work, we address this limitation by exposing the LLM's distribution within the generated programs themselves. We propose a novel test-time framework we dub probabilistic programs of thought to obtain more samples from the model with fewer LLM generations. Given a program generated by a model and the associated next-token probabilities, we build a probabilistic program that compactly represents exponentially many deterministic programs. Since performing probabilistic reasoning in this probabilistic program is much cheaper, our approach allows sampling new programs without any additional GPU compute and little CPU overhead. We instantiate our approach on benchmarks for code generation, code understanding and mathematical reasoning and report improvements in performance with fewer generations from the LLM.

Reinforcement learning with verifiable reward has recently emerged as a central paradigm for post-training large language models (LLMs); however, prevailing mean-based methods, such as Group Relative Policy Optimization (GRPO), suffer from entropy collapse and limited reasoning gains. We argue that these issues stem from overemphasizing high-probability output sequences while neglecting rare but informative reasoning paths. To address these challenges, we propose Risk-based Policy Optimization (RiskPO), which substitutes classical mean-based objectives with principled risk measures. Specifically, we introduce a Mixed Value-at-Risk objective that integrates weighted attention over multiple regions of the reward distribution, thereby amplifying gradient signals on challenging instances and preventing overconfident convergence. We further design a bundling scheme that aggregates multiple questions into bundles, thus enriching the feedback signal and yielding more stable and informative training dynamics. Theoretically, we prove that the risk-averse update alleviates entropy collapse and promotes exploration. Numerically, RiskPO achieves consistent and significant improvements in mathematical reasoning, multi-modal reasoning, and code generation benchmarks, surpassing GRPO and its variants on both Pass@1 and Pass@k metrics. Our results demonstrate that risk-based optimization provides a rigorous and effective paradigm for enhancing LLM reasoning capabilities.

Looped language models separate computational depth from parameter count by repeatedly applying the same transformer block. Adapting these models requires a shared update that remains effective as hidden states evolve throughout the recurrent computation. Our empirical analysis reveals a pronounced late-loop bias in standard low-rank adaptation (LoRA): the shared update is more effective at later loop positions. This imbalance motivates training shared updates under varying combinations of their applications. Randomly omitting adapter applications alone, however, does not improve task performance; it reduces expected update strength during training while leaving inference unchanged. We introduce Loop Dropout, which couples stochastic masking of adapter applications with inverse-survival rescaling to preserve expected update strength and promote effective adaptation across loops. Extensive experiments demonstrate improved mathematical reasoning across model sizes, adapter ranks and training recipes, with benefits extending to general instruction tuning and code generation. Loop Dropout outperforms existing LoRA variants and adapter regularizers, while further analysis shows stronger early-loop adaptation. Every backbone loop remains active, and inference applies the adapter at all loops using standard LoRA without additional trainable parameters or inference computation.

The increasing complexity of AI tasks has shifted the paradigm from monolithic models toward multi-agent large language model (LLM) systems. However, these collaborative architectures introduce a critical bottleneck: redundant prefill computation for shared content generated by previous agents, which significantly increases KV cache memory usage and time-to-first-token (TTFT). While various KV cache methods have been proposed to mitigate prefill redundancy, they either fail to maintain accuracy on agent-generated outputs or exhibit low reuse rates due to rigid constraints. We present RelayCaching, a training-free inference method that directly reuses decoding phase KV caches from previous agents in subsequent prefill phases. Our key insight is that KV caches for identical content are highly consistent across phases, while prefix-induced deviations are sparse and localized within a limited range of layers and token positions. By selectively recomputing KV caches at these positions, RelayCaching preserves model accuracy with minimal overhead, yielding a superior accuracy-efficiency trade-off over existing methods. Experiments on diverse collaborative LLM tasks spanning mathematical reasoning, general knowledge, and code generation demonstrate that RelayCaching achieves over 80% KV cache reuse, reduces TTFT by up to 4.7times compared to the standard pipeline, all with negligible accuracy degradation.

Diffusion large language models promise faster generation by refining many token positions in parallel, but this parallelism introduces a hidden control problem: which proposed tokens should be transferred into the partially decoded sequence at each step? We refer to this decision as token commitment. Existing frozen-generator decoders largely rely on hand-designed confidence rules or block-specific acceptance filters. We argue that token commitment can instead be learned as a reusable trace-state policy. We introduce TraceLock, a lightweight plug-in controller that instantiates this policy for a frozen diffusion language model. Since oracle commitment times are unavailable, TraceLock derives self-supervision from future stability: at decoding step t, a proposed token for position i is labeled stable if it matches the final token at position i after the full decoding trace completes. The controller scores variable-length trace states and decides which active token proposals should be committed to the partially decoded sequence. Once trained for a given frozen backbone, the controller can be deployed across local-window widths, generation lengths, and step budgets without retraining or per-setting calibration. Experiments on question answering, mathematical reasoning, and code generation show that TraceLock improves the quality-step tradeoff over heuristic and learned baselines, with particularly stable behavior under cross-setting deployment. Diagnostic analyses show that its decisions are not reducible to scalar confidence, suggesting that frozen diffusion language models expose a learnable space of commitment trajectories beyond confidence-based decoding. Code is available at https://github.com/BobSun98/TraceLock.

Softmax attention stores key and value vectors for every preceding token, causing inference memory to grow with sequence length. Recent language models incorporating Gated DeltaNet (GDN) or Kimi Delta Attention (KDA) reduce this cost by replacing the KV cache in most layers with fixed-size recurrent states. However, these recurrent states are commonly stored in FP32 and consume substantial GPU memory; their updates are memory-bandwidth bound and contribute significantly to decoding latency. To our knowledge, we are the first to study post-training quantization of recurrent states in GDN and KDA based language models. We find that uniform quantization provides a poor accuracy--storage trade-off: INT8 and FP8 already degrade accuracy on complex reasoning tasks, while INT4 and NVFP4 reduce it to near zero. We further find that most quantization-error energy is concentrated in a small subset of channels and that the relative decay strength of state channels remains stable across prompts and tasks. Motivated by these findings, DAMP uses both quantization-error energy and decay-based persistence to identify high-risk channels during offline calibration. It stores these channels at higher precision and the remainder in INT8. We evaluate DAMP on Qwen3.6-35B and Kimi-Linear-48B across six benchmarks covering mathematical reasoning, general reasoning, and code generation. At 9.9 bits per state value, DAMP maintains average accuracy close to the FP32 baseline. DAMP reduces recurrent-state storage by 69.1%, accelerates the recurrent-state update kernel by up to 2.01x, and lowers full-model TPOT by up to 10.9%.

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.

Recent advances in large language models have led to numerous task-specialized fine-tuned variants, creating a need for efficient model merging techniques that preserve specialized capabilities while avoiding costly retraining. While existing task vector-based merging methods show promise, they typically apply uniform coefficients across all parameters, overlooking varying parameter importance both within and across tasks. We present Sens-Merging, a sensitivity-guided coefficient adjustment method that enhances existing model merging techniques by operating at both task-specific and cross-task levels. Our method analyzes parameter sensitivity within individual tasks and evaluates cross-task transferability to determine optimal merging coefficients. Extensive experiments on Mistral 7B and LLaMA2-7B/13B models demonstrate that Sens-Merging significantly improves performance across general knowledge, mathematical reasoning, and code generation tasks. Notably, when combined with existing merging techniques, our method enables merged models to outperform specialized fine-tuned models, particularly in code generation tasks. Our findings reveal important trade-offs between task-specific and cross-task scalings, providing insights for future model merging strategies.

LLM-based agents can improve themselves across tasks by reusing and revising the skills they orchestrate into executable procedures. Flow-based training fits this loop: it samples procedures in proportion to reward, and the flow through each skill credits it for the next library revision. Three obstacles stand in the way of making this self-improvement reliable: flow training suffers strategy collapse over tree-structured histories; nonnegative flow-based credit rewards frequent use as if it were benefit; and library edits rest on the task reward the policy optimizes. We introduce R^2 Flow, a recursive self-improvement framework that alternates policy learning, independent verification, and versioned skill-library updates on a shared-state orchestration graph. The graph merges histories that differ only in the order of independent steps, allowing flow training to pool evidence across equivalent executions. A flow-share readout of the trained flow, invariant to the backward policy, and a separate signed utility rank which skills to change, verifier evidence decides whether an edit is warranted, and a residual-variance plateau sets when to update. Committed edits reshape the graph the next policy learns on, realizing recursive skill evolution. Across question answering, mathematical reasoning, interactive decision making, and code generation, R^2 Flow improves task accuracy and library-edit precision over heuristic orchestration, reinforcement learning, and skill-evolution baselines, and transfers across executors. Code is available at https://github.com/beita6969/r2flow.

We present Megrez2, a novel lightweight and high-performance language model architecture optimized for device native deployment. Megrez2 introduces a novel cross-layer expert sharing mechanism, which significantly reduces total parameter count by reusing expert modules across adjacent transformer layers while maintaining most of the model's capacity. It also incorporates pre-gated routing, enabling memory-efficient expert loading and faster inference. As the first instantiation of the Megrez2 architecture, we introduce the Megrez2-Preview model, which is pre-trained on a 5-trillion-token corpus and further enhanced through supervised fine-tuning and reinforcement learning with verifiable rewards. With only 3B activated and 7.5B stored parameters, Megrez2-Preview demonstrates competitive or superior performance compared to larger models on a wide range of tasks, including language understanding, instruction following, mathematical reasoning, and code generation. These results highlight the effectiveness of the Megrez2 architecture to achieve a balance between accuracy, efficiency, and deployability, making it a strong candidate for real-world, resource-constrained applications.

Post-training of large language models involves a fundamental trade-off between supervised fine-tuning (SFT), which efficiently mimics demonstrations but tends to memorize, and reinforcement learning (RL), which achieves better generalization at higher computational cost. Dynamic Fine-Tuning (DFT) recently emerged as a promising middle ground, reweighting SFT objectives with token probabilities and achieving improvements in certain reasoning domains, though it exhibits instability in other tasks. We provide a analysis of DFT through the reward-weighted regression (RWR) framework, revealing that it corresponds to a specific auxiliary distribution choice that yields provably tighter RL bounds than standard SFT. However, our analysis also uncovers a critical limitation: this construction lacks distributional anchoring, leading to progressive drift that undermines training stability. To address this, we propose Anchored Supervised Fine-Tuning (ASFT), which augments DFT's reweighting with lightweight KL regularization to preserve tightness while ensuring stability. Empirically, ASFT consistently outperforms both SFT and DFT across mathematical reasoning, medical knowledge grounding, and code generation, achieving substantial improvements with minimal computational overhead. Our RWR framework provides a systematic lens for understanding post-training methods and demonstrates that principled theoretical analysis leads to both stronger guarantees and practical gains.

Diverse outputs in text generation are necessary for effective exploration in complex reasoning tasks, such as code generation and mathematical problem solving. Such Pass@k problems benefit from distinct candidates covering the solution space. However, traditional sampling approaches often waste computational resources on repetitive failure modes. While Diffusion Language Models have emerged as a competitive alternative to the prevailing Autoregressive paradigm, they remain susceptible to this redundancy, with independent samples frequently collapsing into similar modes. To address this, we propose a training free, low cost intervention to enhance generative diversity in Diffusion Language Models. Our approach modifies intermediate samples in a batch sequentially, where each sample is repelled from the feature space of previous samples, actively penalising redundancy. Unlike prior methods that require retraining or beam search, our strategy incurs negligible computational overhead, while ensuring that each sample contributes a unique perspective to the batch. We evaluate our method on the HumanEval and GSM8K benchmarks using the LLaDA-8B-Instruct model. Our results demonstrate significantly improved diversity and Pass@k performance across various temperature settings. As a simple modification to the sampling process, our method offers an immediate, low-cost improvement for current and future Diffusion Language Models in tasks that benefit from diverse solution search. We make our code available at https://github.com/sean-lamont/odd.

Mathematical reasoning remains challenging for LLMs due to complex logic and the need for precise computation. Existing methods enhance LLM reasoning by synthesizing datasets through problem rephrasing, but face issues with generation quality and problem complexity. To address this, we propose to extract structural information with generated problem-solving code from mathematical reasoning and guide data generation with structured solutions. Applied to MATH and GSM8K, our approach produces 39K problems with labeled intermediate steps and a 6.1K-problem benchmark of higher difficulty. Results on our benchmark show that model performance declines as reasoning length increases. Additionally, we conducted fine-tuning experiments using the proposed training data on a range of LLMs, and the results validate the effectiveness of our dataset. We hope the proposed method and dataset will contribute to future research in enhancing LLM reasoning capabilities. Our code and data are available at https://github.com/OpenCausaLab/StructuralGeneration.

Self-Rewarding Language Models propose an architecture in which the Large Language Models(LLMs) both generates responses and evaluates its own outputs via LLM-as-a-Judge prompting, dynamically improving its generative capabilities through iterative Direct Preference Optimization (DPO). However, our analysis reveals a critical limitation in existing Self-Rewarding paradigms: the synchronized improvement of chosen and rejected responses progressively narrows the representational difference between contrasting samples, undermining effective preference learning. We propose Temporal Self-Rewarding Language Models that strategically coordinate past, present, and future model generations to sustain learning signals. Our dual-phase framework introduces: (1) Anchored Rejection - fixing rejected responses using the past initial model's outputs and (2) Future-Guided Chosen - dynamically curating chosen samples using next-generation model predictions. Extensive experiments across three model families (Llama, Qwen, Mistral) and different model sizes (Llama3B/8B/70B) demonstrate significant improvements when trained with our method compared to Self-Rewarding using same computation resources. For example, Llama3.1-8B reaches a 29.44 win rate on AlpacaEval 2.0 with our method, outperforming the Self-Rewarding baseline (19.69) by 9.75. Notably, our method also demonstrates superior out-of-distribution generalization across mathematical reasoning (GSM8K), knowledge-based QA (ARC, TruthfulQA), and code generation (HumanEval) tasks, even though we do not specifically collect such training data.

Large language model (LLM) routers improve the efficiency of multi-model systems by directing each query to the most appropriate model while leveraging the diverse strengths of heterogeneous LLMs. Most existing approaches frame routing as a classification problem based solely on the input query. While this reduces overhead by avoiding inference across all models, it overlooks valuable information that could be gleaned from potential outputs and fails to capture implicit intent or contextual nuances that often emerge only during response generation. These limitations can result in suboptimal routing decisions, particularly for complex or ambiguous queries that require deeper semantic understanding. To address this challenge, we propose Lookahead, a routing framework that "foresees" potential model outputs by predicting their latent representations and uses these predictions to guide model selection, thus enabling more informed routing without full inference. Within this framework, we implement two approaches based on causal and masked language models. Empirical evaluations across seven public benchmarks - spanning instruction following, mathematical reasoning, and code generation - show that Lookahead consistently outperforms existing routing baselines, achieving an average performance gain of 7.7% over the state-of-the-art. Our code is available at https://github.com/huangcb01/lookahead-routing.

Merging Large Language Models (LLMs) aims to amalgamate multiple homologous LLMs into one with all the capabilities. Ideally, any LLMs sharing the same backbone should be mergeable, irrespective of whether they are Fine-Tuned (FT) with minor parameter changes or Pre-Trained (PT) with substantial parameter shifts. However, existing methods often manually assign the model importance, rendering them feasible only for LLMs with similar parameter alterations, such as multiple FT LLMs. The diverse parameter changed ranges between FT and PT LLMs pose challenges for current solutions in empirically determining the optimal combination. In this paper, we make a pioneering effort to broaden the applicability of merging techniques from FT to PT LLMs. We initially examine the efficacy of current methods in merging FT and PT LLMs, discovering that they struggle to deal with PT LLMs. Subsequently, we introduce an approach based on WeIght DisENtanglement (WIDEN) to effectively extend the merging scope, which first disentangles model weights into magnitude and direction components, and then performs adaptive fusion by considering their respective contributions. In the experiments, we merge Qwen1.5-Chat (an FT LLM with instruction-following skills) with Sailor (a PT LLM with multilingual abilities) across 7B and 14B model scales. Results reveal that: (1) existing solutions usually fail when merging Sailor, either losing both abilities or only retaining instruction-following skills; (2) WIDEN successfully injects the multilingual abilities of Sailor into Qwen1.5-Chat and make it proficient in Southeast Asian languages, achieving enhancements in the fundamental capabilities. In light of previous research, we also merge multiple 13B FT LLMs and observe that WIDEN achieves a balanced amalgamation of instruction following, mathematical reasoning, and code generation skills.

Reliable evaluation of large language models (LLMs) is impeded by two key challenges: objective metrics often fail to reflect human perception of natural language, and exhaustive human labeling is prohibitively expensive. Here, we propose a sample-efficient human evaluation method for LLMs based on the principle of MAximum Discrepancy (MAD) Competition. Our method automatically and adaptively selects a compact set of input instructions that maximize semantic discrepancy between pairs of LLM responses. Human evaluators then perform three-alternative forced choices on these paired responses, which are aggregated into a global ranking using Elo rating. We apply our approach to compare eight widely used LLMs across four tasks: scientific knowledge understanding, mathematical reasoning, creative and functional writing, and code generation and explanation. Experimental results show that our sample-efficient evaluation method recovers "gold-standard" model rankings with a handful of MAD-selected instructions, reveals respective strengths and weaknesses of each LLM, and offers nuanced insights to guide future LLM development. Code is available at https://github.com/weiji-Feng/MAD-Eval .

The Muon optimizer has recently offered a promising alternative to AdamW for large language model training, leveraging matrix orthogonalization to produce geometry-aware updates. However, like all first-order methods, Muon can become trapped in sharp local minima. In this work, we present MONA, an optimizer that bridges Muon's orthogonalization framework with curvature-aware acceleration. MONA adds an acceleration term directly into Muon's gradient processing pipeline. This term is calculated from the exponential moving average of gradient differences. We provide a detailed convergence analysis for MONA, showing that the acceleration term enables escape from sharp minima while preserving Muon's spectral-norm regularization. Empirically, MONA achieves better convergence and downstream task performance compared to both Muon and AdamW across three scales of Mixture-of-Experts pretraining, spanning from 1B to 68B parameters, with the largest model trained on 1 trillion tokens. Furthermore, we conduct supervised fine-tuning on the MOE-68B-A3B model and evaluate it on general capability, mathematical reasoning, and code generation benchmarks, where MONA achieves SOTA performance.

Knowledge distillation (KD) transfers knowledge from stronger Teacher models to weaker Student models, but most methods require training the Student parameters, thereby binding the distilled knowledge to a specific architecture and checkpoint. This implicit representation is difficult to interpret or reuse across models and limits KD for API-only or costly-to-train models. This paper studies knowledge transfer for large language models (LLMs). We introduce Universal Textual Teaching (UTT), a parameter-update-free framework that distills observed Teacher-Student knowledge gaps into a textual, interpretable, and reusable natural-language artifact called Primer. Specifically, UTT first identifies representative gap cases through paired evaluations, and iteratively updates the Primer via multi-role interactions: the Student attempts each task, the Prompter turns evaluation feedback into a teaching instruction, the Teacher provides a targeted demonstration, and the Synthesizer consolidates validated lessons. Empirically, on the challenging math (Omni-MATH-2) and code generation (KernelBench) tasks, extensive results confirm the effectiveness of the method: UTT remarkably raises the Student's accuracy from 9.4% to 48.6% and Fast1 accuracy from 9% to 35% on KernelBench, while increasing mathematical reasoning accuracy from 27.6% to 51.7%. UTT also performs better than representative prompt engineering and parameter-based KD methods. Of note, UTT is shown to be generalizable across different Teachers and Students: a Primer synthesized for one Teacher-Student pair can generalize to other Students that do not participate in the synthesis.

Preference distillation typically treats a teacher response as preferred and the student's own response as rejected. This assumes that self-generated failures are the most informative negatives and that rejects must come from a model at least as large as the student, making generation costly at scale. We find neither assumption holds: across students from 7B to 72B, smaller frozen models generate rejects with less inference compute yet train stronger students than self-generated rejects, before and after sequence-level knowledge distillation, on code generation and mathematical reasoning. To explain this result, we derive a finite-horizon utility bound for Direct Preference Optimization in a linearized feature model. The bound characterizes favorable reject distributions and motivates three interventions. First, mixing rejects from smaller and student-scale models improves performance as the smaller model's share increases. Second, reassigning rejects to other prompts and shuffling their code tokens still outperform length-matched gibberish, showing that task structure contributes to reject utility. Third, selecting candidates with lower likelihood under the reference policy improves net transfer when higher-likelihood candidates provide less useful contrast. Lower-likelihood selections outperform higher-likelihood ones for every source. These results suggest that effective rejects preserve task structure while limiting coupling to the reference policy, and that smaller frozen models can provide them at low cost.

Large Language Model (LLM) based multi-agent systems have shown remarkable performance in various tasks, especially when enhanced through collaborative communication. However, current methods often rely on a fixed number of agents and static communication structures, limiting their ability to adapt to varying task complexities. In this paper, we propose Adaptive Graph Pruning (AGP), a novel task-adaptive multi-agent collaboration framework that jointly optimizes agent quantity (hard-pruning) and communication topology (soft-pruning). Specifically, our method employs a two-stage training strategy: firstly, independently training soft-pruning networks for different agent quantities to determine optimal agent-quantity-specific complete graphs and positional masks across specific tasks; and then jointly optimizing hard-pruning and soft-pruning within a maximum complete graph to dynamically configure the number of agents and their communication topologies per task. Extensive experiments demonstrate that our approach is: (1) High-performing, achieving state-of-the-art results across six benchmarks and consistently generalizes across multiple mainstream LLM architectures, with a increase in performance of 2.58%sim 9.84%; (2) Task-adaptive, dynamically constructing optimized communication topologies tailored to specific tasks, with an extremely high performance in all three task categories (general reasoning, mathematical reasoning, and code generation); (3) Token-economical, having fewer training steps and token consumption at the same time, with a decrease in token consumption of 90%+; and (4) Training-efficient, achieving high performance with very few training steps compared with other methods. The performance will surpass the existing baselines after about ten steps of training under six benchmarks.

Self-correction of large language models (LLMs) emerges as a critical component for enhancing their reasoning performance. Although various self-correction methods have been proposed, a comprehensive evaluation of these methods remains largely unexplored, and the question of whether LLMs can truly correct themselves is a matter of significant interest and concern. In this study, we introduce CorrectBench, a benchmark developed to evaluate the effectiveness of self-correction strategies, including intrinsic, external, and fine-tuned approaches, across three tasks: commonsense reasoning, mathematical reasoning, and code generation. Our findings reveal that: 1) Self-correction methods can improve accuracy, especially for complex reasoning tasks; 2) Mixing different self-correction strategies yields further improvements, though it reduces efficiency; 3) Reasoning LLMs (e.g., DeepSeek-R1) have limited optimization under additional self-correction methods and have high time costs. Interestingly, a comparatively simple chain-of-thought (CoT) baseline demonstrates competitive accuracy and efficiency. These results underscore the potential of self-correction to enhance LLM's reasoning performance while highlighting the ongoing challenge of improving their efficiency. Consequently, we advocate for further research focused on optimizing the balance between reasoning capabilities and operational efficiency. Project Page: https://correctbench.github.io/

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.

Multimodal large language models (MLLMs) have demonstrated remarkable capabilities in vision-language answering tasks. Despite their strengths, these models often encounter challenges in achieving complex reasoning tasks such as mathematical problem-solving. Previous works have focused on fine-tuning on specialized mathematical datasets. However, these datasets are typically distilled directly from teacher models, which capture only static reasoning patterns and leaving substantial gaps compared to student models. This reliance on fixed teacher-derived datasets not only restricts the model's ability to adapt to novel or more intricate questions that extend beyond the confines of the training data, but also lacks the iterative depth needed for robust generalization. To overcome these limitations, we propose \method, a Mathematical Self-Evolving framework for MLLMs. In contrast to traditional one-shot fine-tuning paradigms, \method iteratively refines the model through cycles of inference, reflection, and reward-based feedback. Specifically, we leverage iterative fine-tuning by incorporating correct reasoning paths derived from previous-stage inference and integrating reflections from a specialized Outcome Reward Model (ORM). To verify the effectiveness of \method, we evaluate it on a suite of challenging benchmarks, demonstrating significant performance gains over backbone models. Notably, our experimental results on MathVL-test surpass the leading open-source multimodal mathematical reasoning model QVQ. Our code and models are available at https://zheny2751\allowbreak-dotcom.github.io/\allowbreak MathSE.github.io/.

Reinforcement Learning with Verifiable Rewards (RLVR) offers a robust mechanism for enhancing mathematical reasoning in large models. However, we identify a systematic lack of emphasis on more challenging questions in existing methods from both algorithmic and data perspectives, despite their importance for refining underdeveloped capabilities. Algorithmically, widely used Group Relative Policy Optimization (GRPO) suffers from an implicit imbalance where the magnitude of policy updates is lower for harder questions. Data-wise, augmentation approaches primarily rephrase questions to enhance diversity without systematically increasing intrinsic difficulty. To address these issues, we propose a two-dual MathForge framework to improve mathematical reasoning by targeting harder questions from both perspectives, which comprises a Difficulty-Aware Group Policy Optimization (DGPO) algorithm and a Multi-Aspect Question Reformulation (MQR) strategy. Specifically, DGPO first rectifies the implicit imbalance in GRPO via difficulty-balanced group advantage estimation, and further prioritizes harder questions by difficulty-aware question-level weighting. Meanwhile, MQR reformulates questions across multiple aspects to increase difficulty while maintaining the original gold answer. Overall, MathForge forms a synergistic loop: MQR expands the data frontier, and DGPO effectively learns from the augmented data. Extensive experiments show that MathForge significantly outperforms existing methods on various mathematical reasoning tasks. The code and augmented data are all available at https://github.com/AMAP-ML/MathForge.

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 .

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.

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.

Large language models have demonstrated remarkable capabilities in complex mathematical reasoning tasks, but they inevitably generate errors throughout multi-step solutions. Process-level Reward Models (PRMs) have shown great promise by providing supervision and evaluation at each intermediate step, thereby effectively improving the models' reasoning abilities. However, training effective PRMs requires high-quality process reward data, yet existing methods for constructing such data are often labour-intensive or inefficient. In this paper, we propose an uncertainty-driven framework for automated process reward data construction, encompassing both data generation and annotation processes for PRMs. Additionally, we identify the limitations of both majority vote and PRMs, and introduce two generic uncertainty-aware output aggregation methods: Hybrid Majority Reward Vote and Weighted Reward Frequency Vote, which combine the strengths of majority vote with PRMs. Extensive experiments on ProcessBench, MATH, and GSMPlus show the effectiveness and efficiency of the proposed PRM data construction framework, and demonstrate that the two output aggregation methods further improve the mathematical reasoning abilities across diverse PRMs. The code and data will be publicly available at https://github.com/Jiuzhouh/UnPRM.

Despite the strong performance of large language models (LLMs) in tasks like mathematical reasoning, their practical use is limited by high computational demands and proprietary restrictions. Chain-of-thought (CoT) and program-of-thought (PoT) fine-tuning are common methods to transfer LLM knowledge to small language models (SLMs). However, CoT often leads to calculation errors in SLMs, while PoT has shown more promise. While most PoT-based approaches focus on direct problem-to-code conversion or extracting only the key information from questions and then providing code solution for it, this work emphasizes filling the gaps in the question to clearly illustrate the solution path, which can be challenging for an SLM to understand when such information is not explicitly provided. Therefore, this paper introduces Gap-Filling Prompting (GFP), a novel two-step prompting strategy designed to enhance the problem-solving process for SLMs. The first step identifies these gaps and provides hints for filling them, while the second step adds the hints to the question to generate a final code solution. Experimental results on two benchmark datasets demonstrate that GFP significantly improves the mathematical reasoning abilities of SLMs.

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.

Mathematical reasoning is an important capability of large language models~(LLMs) for real-world applications. To enhance this capability, existing work either collects large-scale math-related texts for pre-training, or relies on stronger LLMs (\eg GPT-4) to synthesize massive math problems. Both types of work generally lead to large costs in training or synthesis. To reduce the cost, based on open-source available texts, we propose an efficient way that trains a small LLM for math problem synthesis, to efficiently generate sufficient high-quality pre-training data. To achieve it, we create a dataset using GPT-4 to distill its data synthesis capability into the small LLM. Concretely, we craft a set of prompts based on human education stages to guide GPT-4, to synthesize problems covering diverse math knowledge and difficulty levels. Besides, we adopt the gradient-based influence estimation method to select the most valuable math-related texts. The both are fed into GPT-4 for creating the knowledge distillation dataset to train the small LLM. We leverage it to synthesize 6 million math problems for pre-training our JiuZhang3.0 model, which only needs to invoke GPT-4 API 9.3k times and pre-train on 4.6B data. Experimental results have shown that JiuZhang3.0 achieves state-of-the-art performance on several mathematical reasoning datasets, under both natural language reasoning and tool manipulation settings. Our code and data will be publicly released in https://github.com/RUCAIBox/JiuZhang3.0.

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.

Mathematics has long been conveyed through natural language, primarily for human understanding. With the rise of mechanized mathematics and proof assistants, there is a growing need to understand informal mathematical text, yet most existing benchmarks focus solely on English, overlooking other languages. This paper introduces RoMath, a Romanian mathematical reasoning benchmark suite comprising three datasets: RoMath-Baccalaureate, RoMath-Competitions and RoMath-Synthetic, which cover a range of mathematical domains and difficulty levels, aiming to improve non-English language models and promote multilingual AI development. By focusing on Romanian, a low-resource language with unique linguistic features, RoMath addresses the limitations of Anglo-centric models and emphasizes the need for dedicated resources beyond simple automatic translation. We benchmark several open-weight language models, highlighting the importance of creating resources for underrepresented languages. We make the code and dataset available.