Parameter-efficient fine-tuning (PEFT) methods have shown promise in adapting large language models, yet existing approaches exhibit counter-intuitive phenomena: integrating router into prompt tuning (PT) increases training efficiency yet does not improve performance universally; parameter reduction through matrix decomposition can improve performance in specific domains. Motivated by these observations and the modular nature of PT, we propose PT-MoE, a novel framework that integrates matrix decomposition with mixture-of-experts (MoE) routing for efficient PT. Results across 17 datasets demonstrate that PT-MoE achieves state-of-the-art performance in both question answering (QA) and mathematical problem solving tasks, improving F1 score by 1.49 points over PT and 2.13 points over LoRA in QA tasks, while enhancing mathematical accuracy by 10.75 points over PT and 0.44 points over LoRA, all while using 25% fewer parameters than LoRA. Our analysis reveals that while PT methods generally excel in QA tasks and LoRA-based methods in math datasets, the integration of matrix decomposition and MoE in PT-MoE yields complementary benefits: decomposition enables efficient parameter sharing across experts while MoE provides dynamic adaptation, collectively enabling PT-MoE to demonstrate cross-task consistency and generalization abilities. These findings, along with ablation studies on routing mechanisms and architectural components, provide insights for future PEFT methods.
Large Language Models (LLMs) often struggle with mathematical reasoning tasks requiring precise, verifiable computation. While Reinforcement Learning (RL) from outcome-based rewards enhances text-based reasoning, understanding how agents autonomously learn to leverage external tools like code execution remains crucial. We investigate RL from outcome-based rewards for Tool-Integrated Reasoning, ZeroTIR, training base LLMs to spontaneously generate and execute Python code for mathematical problems without supervised tool-use examples. Our central contribution is we demonstrate that as RL training progresses, key metrics scale predictably. Specifically, we observe strong positive correlations where increased training steps lead to increases in the spontaneous code execution frequency, the average response length, and, critically, the final task accuracy. This suggests a quantifiable relationship between computational effort invested in training and the emergence of effective, tool-augmented reasoning strategies. We implement a robust framework featuring a decoupled code execution environment and validate our findings across standard RL algorithms and frameworks. Experiments show ZeroTIR significantly surpasses non-tool ZeroRL baselines on challenging math benchmarks. Our findings provide a foundational understanding of how autonomous tool use is acquired and scales within Agent RL, offering a reproducible benchmark for future studies. Code is released at \href{https://github.com/yyht/openrlhf_async_pipline}{https://github.com/yyht/openrlhf\_async\_pipline}.
Reasoning capabilities have significantly improved the performance of vision-language models (VLMs) in domains such as mathematical problem-solving, coding, and visual question-answering. However, their impact on real-world applications remains unclear. This paper presents the first empirical study on the effectiveness of reasoning-enabled VLMs in mobile GUI agents, a domain that requires interpreting complex screen layouts, understanding user instructions, and executing multi-turn interactions. We evaluate two pairs of commercial models--Gemini 2.0 Flash and Claude 3.7 Sonnet--comparing their base and reasoning-enhanced versions across two static benchmarks (ScreenSpot and AndroidControl) and one interactive environment (AndroidWorld). We surprisingly find the Claude 3.7 Sonnet reasoning model achieves state-of-the-art performance on AndroidWorld. However, reasoning VLMs generally offer marginal improvements over non-reasoning models on static benchmarks and even degrade performance in some agent setups. Notably, reasoning and non-reasoning VLMs fail on different sets of tasks, suggesting that reasoning does have an impact, but its benefits and drawbacks counterbalance each other. We attribute these inconsistencies to the limitations of benchmarks and VLMs. Based on the findings, we provide insights for further enhancing mobile GUI agents in terms of benchmarks, VLMs, and their adaptability in dynamically invoking reasoning VLMs. The experimental data are publicly available at https://github.com/LlamaTouch/VLM-Reasoning-Traces.
Despite impressive performance across diverse tasks, Multimodal Large Language Models (MLLMs) have yet to fully demonstrate their potential in visual mathematical problem-solving, particularly in accurately perceiving and interpreting diagrams. Inspired by typical processes of humans, we hypothesize that the perception capabilities to extract meaningful information from diagrams is crucial, as it directly impacts subsequent inference processes. To validate this hypothesis, we developed FlowVerse, a comprehensive benchmark that categorizes all information used during problem-solving into four components, which are then combined into six problem versions for evaluation. Our preliminary results on FlowVerse reveal that existing MLLMs exhibit substantial limitations when extracting essential information and reasoned property from diagrams and performing complex reasoning based on these visual inputs. In response, we introduce MathFlow, a modular problem-solving pipeline that decouples perception and inference into distinct stages, thereby optimizing each independently. Given the perceptual limitations observed in current MLLMs, we trained MathFlow-P-7B as a dedicated perception model. Experimental results indicate that MathFlow-P-7B yields substantial performance gains when integrated with various closed-source and open-source inference models. This demonstrates the effectiveness of the MathFlow pipeline and its compatibility to diverse inference frameworks. The FlowVerse benchmark and code are available at https://github.com/MathFlow-zju/MathFlow.
Large language models (LLMs) have significantly improved their reasoning capabilities; however, they still struggle with complex multi-step mathematical problem-solving due to error propagation, lack of self-correction, and limited adaptability to diverse reasoning styles. Existing methods rely on static fine-tuning or prompt engineering, which fail to generalize across problem complexities, while the scarcity of high-quality preference data further hinders reliable reasoning. We introduce SPHERE, a self-evolving data generation pipeline that enhances reasoning in small language models (SLMs) by iteratively generating, correcting, and diversifying reasoning chains. SPHERE operates in three stages: (i) Self-Generation, where the model autonomously constructs problem-solving steps; (ii) Self-Correction, enabling it to identify and rectify errors; and (iii) Diversity Induction, improving robustness through multiple valid reasoning trajectories. This self-evolution mechanism strengthens mathematical reasoning and enhances model reliability. Evaluations on MATH 500, GSM8K, AIME, AMC, and Olympiad show that SPHERE-trained models achieve significant gains over their base versions and match/surpass GPT-4o on certain benchmarks. Our findings demonstrate that self-evolving models can close the reasoning gap between SLMs and state-of-the-art LLMs, making mathematical AI more reliable, scalable, and efficient.
Recent LLMs have significantly improved reasoning capabilities, primarily by including an explicit, lengthy Thinking process as part of generation. In this paper, we question whether this explicit thinking is necessary. Using the state-of-the-art DeepSeek-R1-Distill-Qwen, we find that bypassing the thinking process via simple prompting, denoted as NoThinking, can be surprisingly effective. When controlling for the number of tokens, NoThinking outperforms Thinking across a diverse set of seven challenging reasoning datasets--including mathematical problem solving, formal theorem proving, and coding--especially in low-budget settings, e.g., 51.3 vs. 28.9 on ACM 23 with 700 tokens. Notably, the performance of NoThinking becomes more competitive with pass@k as k increases. Building on this observation, we demonstrate that a parallel scaling approach that uses NoThinking to generate N outputs independently and aggregates them is highly effective. For aggregation, we use task-specific verifiers when available, or we apply simple best-of-N strategies such as confidence-based selection. Our method outperforms a range of baselines with similar latency using Thinking, and is comparable to Thinking with significantly longer latency (up to 9x). Together, our research encourages a reconsideration of the necessity of lengthy thinking processes, while also establishing a competitive reference for achieving strong reasoning performance in low-budget settings or at low latency using parallel scaling.
This technical report presents Ring-Lite-Distill, a lightweight reasoning model derived from our open-source Mixture-of-Experts (MoE) Large Language Models (LLMs) Ling-Lite. This study demonstrates that through meticulous high-quality data curation and ingenious training paradigms, the compact MoE model Ling-Lite can be further trained to achieve exceptional reasoning capabilities, while maintaining its parameter-efficient architecture with only 2.75 billion activated parameters, establishing an efficient lightweight reasoning architecture. In particular, in constructing this model, we have not merely focused on enhancing advanced reasoning capabilities, exemplified by high-difficulty mathematical problem solving, but rather aimed to develop a reasoning model with more comprehensive competency coverage. Our approach ensures coverage across reasoning tasks of varying difficulty levels while preserving generic capabilities, such as instruction following, tool use, and knowledge retention. We show that, Ring-Lite-Distill's reasoning ability reaches a level comparable to DeepSeek-R1-Distill-Qwen-7B, while its general capabilities significantly surpass those of DeepSeek-R1-Distill-Qwen-7B. The models are accessible at https://huggingface.co/inclusionAI
It is a widely known issue that Transformers, when trained on shorter sequences, fail to generalize robustly to longer ones at test time. This raises the question of whether Transformer models are real reasoning engines, despite their impressive abilities in mathematical problem solving and code synthesis. In this paper, we offer a vanishing variance perspective on this issue. To the best of our knowledge, we are the first to demonstrate that even for today's frontier models, a longer sequence length results in a decrease in variance in the output of the multi-head attention modules. On the argmax retrieval and dictionary lookup tasks, our experiments show that applying layer normalization after the attention outputs leads to significantly better length generalization. Our analyses attribute this improvement to a reduction-though not a complete elimination-of the distribution shift caused by vanishing variance.
Natural Language to SQL (NL2SQL) has emerged as a critical task for enabling seamless interaction with databases. Recent advancements in Large Language Models (LLMs) have demonstrated remarkable performance in this domain. However, existing NL2SQL methods predominantly rely on closed-source LLMs leveraging prompt engineering, while open-source models typically require fine-tuning to acquire domain-specific knowledge. Despite these efforts, open-source LLMs struggle with complex NL2SQL tasks due to the indirect expression of user query objectives and the semantic gap between user queries and database schemas. Inspired by the application of reinforcement learning in mathematical problem-solving to encourage step-by-step reasoning in LLMs, we propose LearNAT (Learning NL2SQL with AST-guided Task Decomposition), a novel framework that improves the performance of open-source LLMs on complex NL2SQL tasks through task decomposition and reinforcement learning. LearNAT introduces three key components: (1) a Decomposition Synthesis Procedure that leverages Abstract Syntax Trees (ASTs) to guide efficient search and pruning strategies for task decomposition, (2) Margin-aware Reinforcement Learning, which employs fine-grained step-level optimization via DPO with AST margins, and (3) Adaptive Demonstration Reasoning, a mechanism for dynamically selecting relevant examples to enhance decomposition capabilities. Extensive experiments on two benchmark datasets, Spider and BIRD, demonstrate that LearNAT enables a 7B-parameter open-source LLM to achieve performance comparable to GPT-4, while offering improved efficiency and accessibility.
This study investigates the reasoning robustness of large language models (LLMs) on mathematical problem-solving tasks under systematically introduced input perturbations. Using the GSM8K dataset as a controlled testbed, we evaluate how well state-of-the-art models maintain logical consistency and correctness when confronted with four categories of prompt perturbations: irrelevant context, pathological instructions, factually relevant but non-essential context, and a combination of the latter two. Our experiments, conducted on thirteen open-source and closed-source LLMs, reveal that introducing irrelevant context within the model's context window significantly degrades performance, suggesting that distinguishing essential from extraneous details remains a pressing challenge. Surprisingly, performance regressions are relatively insensitive to the complexity of the reasoning task, as measured by the number of steps required, and are not strictly correlated with model size. Moreover, we observe that certain perturbations inadvertently trigger chain-of-thought-like reasoning behaviors, even without explicit prompting. Our findings highlight critical vulnerabilities in current LLMs and underscore the need for improved robustness against noisy, misleading, and contextually dense inputs, paving the way for more resilient and reliable reasoning in real-world applications.
Recent advancements in large language models (LLMs) have shown impressive progress in mathematical reasoning tasks. However, current evaluation benchmarks predominantly focus on the accuracy of final answers, often overlooking the logical rigor crucial for mathematical problem-solving. The claim that state-of-the-art LLMs can solve Math Olympiad-level problems requires closer examination. To explore this, we conducted both qualitative and quantitative human evaluations of proofs generated by LLMs, and developed a schema for automatically assessing their reasoning capabilities. Our study reveals that current LLMs fall significantly short of solving challenging Olympiad-level problems and frequently fail to distinguish correct mathematical reasoning from clearly flawed solutions. We also found that occasional correct final answers provided by LLMs often result from pattern recognition or heuristic shortcuts rather than genuine mathematical reasoning. These findings underscore the substantial gap between LLM performance and human expertise in advanced mathematical reasoning and highlight the importance of developing benchmarks that prioritize the rigor and coherence of mathematical arguments rather than merely the correctness of final answers.
The mathematical problem-solving capabilities of large language models have become a focal point of research, with growing interests in leveraging self-generated reasoning paths as a promising way to refine and enhance these models. These paths capture step-by-step logical processes while requiring only the correct answer for supervision. The self-training method has been shown to be effective in reasoning tasks while eliminating the need for external models and manual annotations. However, optimizing the use of self-generated data for model training remains an open challenge. In this work, we propose Entropy-Based Adaptive Weighting for Self-Training (EAST), an adaptive weighting strategy designed to prioritize uncertain data during self-training. Specifically, EAST employs a mapping function with a tunable parameter that controls the sharpness of the weighting, assigning higher weights to data where the model exhibits greater uncertainty. This approach guides the model to focus on more informative and challenging examples, thereby enhancing its reasoning ability. We evaluate our approach on GSM8K and MATH benchmarks. Empirical results show that, while the vanilla method yields virtually no improvement (0%) on MATH, EAST achieves around a 1% gain over backbone model. On GSM8K, EAST attains a further 1-2% performance boost compared to the vanilla method.
Mathematical error detection in educational settings presents a significant challenge for Multimodal Large Language Models (MLLMs), requiring a sophisticated understanding of both visual and textual mathematical content along with complex reasoning capabilities. Though effective in mathematical problem-solving, MLLMs often struggle with the nuanced task of identifying and categorizing student errors in multimodal mathematical contexts. Therefore, we introduce MathAgent, a novel Mixture-of-Math-Agent framework designed specifically to address these challenges. Our approach decomposes error detection into three phases, each handled by a specialized agent: an image-text consistency validator, a visual semantic interpreter, and an integrative error analyzer. This architecture enables more accurate processing of mathematical content by explicitly modeling relationships between multimodal problems and student solution steps. We evaluate MathAgent on real-world educational data, demonstrating approximately 5% higher accuracy in error step identification and 3% improvement in error categorization compared to baseline models. Besides, MathAgent has been successfully deployed in an educational platform that has served over one million K-12 students, achieving nearly 90% student satisfaction while generating significant cost savings by reducing manual error detection.
Mathematical reasoning and optimization are fundamental to artificial intelligence and computational problem-solving. Recent advancements in Large Language Models (LLMs) have significantly improved AI-driven mathematical reasoning, theorem proving, and optimization techniques. This survey explores the evolution of mathematical problem-solving in AI, from early statistical learning approaches to modern deep learning and transformer-based methodologies. We review the capabilities of pretrained language models and LLMs in performing arithmetic operations, complex reasoning, theorem proving, and structured symbolic computation. A key focus is on how LLMs integrate with optimization and control frameworks, including mixed-integer programming, linear quadratic control, and multi-agent optimization strategies. We examine how LLMs assist in problem formulation, constraint generation, and heuristic search, bridging theoretical reasoning with practical applications. We also discuss enhancement techniques such as Chain-of-Thought reasoning, instruction tuning, and tool-augmented methods that improve LLM's problem-solving performance. Despite their progress, LLMs face challenges in numerical precision, logical consistency, and proof verification. Emerging trends such as hybrid neural-symbolic reasoning, structured prompt engineering, and multi-step self-correction aim to overcome these limitations. Future research should focus on interpretability, integration with domain-specific solvers, and improving the robustness of AI-driven decision-making. This survey offers a comprehensive review of the current landscape and future directions of mathematical reasoning and optimization with LLMs, with applications across engineering, finance, and scientific research.
Large Language Models (LLMs) have shown impressive progress in mathematical reasoning. While data augmentation is promising to enhance mathematical problem-solving ability, current approaches are predominantly limited to instance-level modifications-such as rephrasing or generating syntactic variations-which fail to capture and leverage the intrinsic relational structures inherent in mathematical knowledge. Inspired by human learning processes, where mathematical proficiency develops through systematic exposure to interconnected concepts, we introduce MathFusion, a novel framework that enhances mathematical reasoning through cross-problem instruction synthesis. MathFusion implements this through three fusion strategies: (1) sequential fusion, which chains related problems to model solution dependencies; (2) parallel fusion, which combines analogous problems to reinforce conceptual understanding; and (3) conditional fusion, which creates context-aware selective problems to enhance reasoning flexibility. By applying these strategies, we generate a new dataset, \textbf{MathFusionQA}, followed by fine-tuning models (DeepSeekMath-7B, Mistral-7B, Llama3-8B) on it. Experimental results demonstrate that MathFusion achieves substantial improvements in mathematical reasoning while maintaining high data efficiency, boosting performance by 18.0 points in accuracy across diverse benchmarks while requiring only 45K additional synthetic instructions, representing a substantial improvement over traditional single-instruction approaches. Our datasets, models, and code are publicly available at https://github.com/QizhiPei/mathfusion.
This paper presents an in-depth analysis of the performance of seven different Large Language Models (LLMs) in solving a diverse set of math advanced calculus problems. The study aims to evaluate these models' accuracy, reliability, and problem-solving capabilities, including ChatGPT 4o, Gemini Advanced with 1.5 Pro, Copilot Pro, Claude 3.5 Sonnet, Meta AI, Mistral AI, and Perplexity. The assessment was conducted through a series of thirty-two test problems, encompassing a total of 320 points. The problems covered various topics, from vector calculations and geometric interpretations to integral evaluations and optimization tasks. The results highlight significant trends and patterns in the models' performance, revealing both their strengths and weaknesses - for instance, models like ChatGPT 4o and Mistral AI demonstrated consistent accuracy across various problem types, indicating their robustness and reliability in mathematical problem-solving, while models such as Gemini Advanced with 1.5 Pro and Meta AI exhibited specific weaknesses, particularly in complex problems involving integrals and optimization, suggesting areas for targeted improvements. The study also underscores the importance of re-prompting in achieving accurate solutions, as seen in several instances where models initially provided incorrect answers but corrected them upon re-prompting. Overall, this research provides valuable insights into the current capabilities and limitations of LLMs in the domain of math calculus, with the detailed analysis of each model's performance on specific problems offering a comprehensive understanding of their strengths and areas for improvement, contributing to the ongoing development and refinement of LLM technology. The findings are particularly relevant for educators, researchers, and developers seeking to leverage LLMs for educational and practical applications in mathematics.
Recent advancements in Large Language Models (LLMs) have substantially evolved Multi-Agent Systems (MASs) capabilities, enabling systems that not only automate tasks but also leverage near-human reasoning capabilities. To achieve this, LLM-based MASs need to be built around two critical principles: (i) a robust architecture that fully exploits LLM potential for specific tasks -- or related task sets -- and ($ii$) an effective methodology for equipping LLMs with the necessary capabilities to perform tasks and manage information efficiently. It goes without saying that a priori architectural designs can limit the scalability and domain adaptability of a given MAS. To address these challenges, in this paper we introduce Nexus: a lightweight Python framework designed to easily build and manage LLM-based MASs. Nexus introduces the following innovations: (i) a flexible multi-supervisor hierarchy, (ii) a simplified workflow design, and (iii) easy installation and open-source flexibility: Nexus can be installed via pip and is distributed under a permissive open-source license, allowing users to freely modify and extend its capabilities. Experimental results demonstrate that architectures built with Nexus exhibit state-of-the-art performance across diverse domains. In coding tasks, Nexus-driven MASs achieve a 99% pass rate on HumanEval and a flawless 100% on VerilogEval-Human, outperforming cutting-edge reasoning language models such as o3-mini and DeepSeek-R1. Moreover, these architectures display robust proficiency in complex reasoning and mathematical problem solving, achieving correct solutions for all randomly selected problems from the MATH dataset. In the realm of multi-objective optimization, Nexus-based architectures successfully address challenging timing closure tasks on designs from the VTR benchmark suite, while guaranteeing, on average, a power saving of nearly 30%.
Recent research across mathematical problem solving, proof assistant programming and multimodal jailbreaking documents a striking finding: when (multimodal) language model tackle a suite of tasks with multiple attempts per task -- succeeding if any attempt is correct -- then the negative log of the average success rate scales a power law in the number of attempts. In this work, we identify an apparent puzzle: a simple mathematical calculation predicts that on each problem, the failure rate should fall exponentially with the number of attempts. We confirm this prediction empirically, raising a question: from where does aggregate polynomial scaling emerge? We then answer this question by demonstrating per-problem exponential scaling can be made consistent with aggregate polynomial scaling if the distribution of single-attempt success probabilities is heavy tailed such that a small fraction of tasks with extremely low success probabilities collectively warp the aggregate success trend into a power law - even as each problem scales exponentially on its own. We further demonstrate that this distributional perspective explains previously observed deviations from power law scaling, and provides a simple method for forecasting the power law exponent with an order of magnitude lower relative error, or equivalently, ${\sim}2-4$ orders of magnitude less inference compute. Overall, our work contributes to a better understanding of how neural language model performance improves with scaling inference compute and the development of scaling-predictable evaluations of (multimodal) language models.
With the rapid development of large language models (LLMs), fully fine-tuning (FT) these models has become increasingly impractical due to the high computational demands. Additionally, FT can lead to catastrophic forgetting. As an alternative, Low-Rank Adaptation (LoRA) has been proposed, which fine-tunes only a small subset of parameters, achieving similar performance to FT while significantly reducing resource requirements. However, since LoRA inherits FT's design, the issue of catastrophic forgetting remains. To address these challenges, we propose SECURA: Sigmoid-Enhanced CUR Decomposition LoRA, a novel parameter-efficient fine-tuning (PEFT) variant that mitigates catastrophic forgetting while improving fine-tuning performance. Our method introduces a new normalization technique, SigNorm, to enhance parameter retention and overall performance. SECURA has been evaluated on a variety of tasks, including mathematical problem-solving (GSM8K), challenging question-answering (CNNDM), translation (NewsDE), and complex multiple-choice reasoning (LogiQA). Experimental results show that SECURA achieves an average fine-tuning improvement of 3.59% across four multiple-choice question (MCQ) tasks and a 2.51% improvement across five question-answering (QA) tasks on models such as Gemma2 2b, Qwen2 1.5b, Qwen 2 7b, Llama3 8b, and Llama3.1 8b, compared to DoRA. Moreover, SECURA demonstrates superior knowledge retention capabilities, maintaining more than 70% accuracy on basic LLM knowledge across 16 continual learning tests, outperforming Experience Replay (ER), Sequential Learning (SEQ), EWC, I-LoRA, and CUR-LoRA.
Despite strong performance on vision-language tasks, Multimodal Large Language Models (MLLMs) struggle with mathematical problem-solving, with both open-source and state-of-the-art models falling short of human performance on visual-math benchmarks. To systematically examine visual-mathematical reasoning in MLLMs, we (1) evaluate their understanding of geometric primitives, (2) test multi-step reasoning, and (3) explore a potential solution to improve visual reasoning capabilities. Our findings reveal fundamental shortcomings in shape recognition, with top models achieving under 50% accuracy in identifying regular polygons. We analyze these failures through the lens of dual-process theory and show that MLLMs rely on System 1 (intuitive, memorized associations) rather than System 2 (deliberate reasoning). Consequently, MLLMs fail to count the sides of both familiar and novel shapes, suggesting they have neither learned the concept of sides nor effectively process visual inputs. Finally, we propose Visually Cued Chain-of-Thought (VC-CoT) prompting, which enhances multi-step mathematical reasoning by explicitly referencing visual annotations in diagrams, boosting GPT-4o's accuracy on an irregular polygon side-counting task from 7% to 93%. Our findings suggest that System 2 reasoning in MLLMs remains an open problem, and visually-guided prompting is essential for successfully engaging visual reasoning. Code available at: https://github.com/rsinghlab/Shape-Blind.