High-quality test items are essential for educational assessments, particularly within Item Response Theory (IRT). Traditional validation methods rely on resource-intensive pilot testing to estimate item difficulty and discrimination. More recently, Item-Writing Flaw (IWF) rubrics emerged as a domain-general approach for evaluating test items based on textual features. However, their relationship to IRT parameters remains underexplored. To address this gap, we conducted a study involving over 7,000 multiple-choice questions across various STEM subjects (e.g., math and biology). Using an automated approach, we annotated each question with a 19-criteria IWF rubric and studied relationships to data-driven IRT parameters. Our analysis revealed statistically significant links between the number of IWFs and IRT difficulty and discrimination parameters, particularly in life and physical science domains. We further observed how specific IWF criteria can impact item quality more and less severely (e.g., negative wording vs. implausible distractors). Overall, while IWFs are useful for predicting IRT parameters--particularly for screening low-difficulty MCQs--they cannot replace traditional data-driven validation methods. Our findings highlight the need for further research on domain-general evaluation rubrics and algorithms that understand domain-specific content for robust item validation.
An ideal model evaluation should achieve two goals: identifying where the model fails and providing actionable improvement guidance. Toward these goals for Language Model (LM) evaluations, we formulate the problem of generating a weakness profile, a set of weaknesses expressed in natural language, given an LM's performance on every individual instance in a benchmark. We introduce a suite of quantitative assessments to compare different weakness profiling methods. We also propose a weakness profiling method EvalTree. It constructs a capability tree where each node represents a capability described in natural language and is linked to a subset of benchmark instances that specifically evaluate this capability; it then extracts nodes where the LM performs poorly to generate a weakness profile. On the MATH and WildChat benchmarks, we show that EvalTree outperforms baseline weakness profiling methods by identifying weaknesses more precisely and comprehensively. Weakness profiling further enables weakness-guided data collection, and training data collection guided by EvalTree-identified weaknesses improves LM performance more than other data collection strategies. We also show how EvalTree exposes flaws in Chatbot Arena's human-voter-based evaluation practice. To facilitate future work, we release our code and an interface that allows practitioners to interactively explore the capability trees built by EvalTree.
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.
Combining existing pre-trained expert LLMs is a promising avenue for scalably tackling large-scale and diverse tasks. However, selecting experts at the task level is often too coarse-grained, as heterogeneous tasks may require different expertise for each instance. To enable adaptive instance-level mixing of pre-trained LLM experts, we propose Symbolic-MoE, a symbolic, text-based, and gradient-free Mixture-of-Experts framework. Symbolic-MoE takes a fine-grained approach to selection by emphasizing skills, e.g., algebra in math or molecular biology in biomedical reasoning. We propose a skill-based recruiting strategy that dynamically selects the most relevant set of expert LLMs for diverse reasoning tasks based on their strengths. Each selected expert then generates its own reasoning, resulting in k outputs from k experts, which are then synthesized into a final high-quality response by an aggregator chosen based on its ability to integrate diverse reasoning outputs. We show that Symbolic-MoE's instance-level expert selection improves performance by a large margin but -- when implemented naively -- can introduce a high computational overhead due to the need for constant model loading and offloading. To address this, we implement a batch inference strategy that groups instances based on their assigned experts, loading each model only once. This allows us to integrate 16 expert models on 1 GPU with a time cost comparable to or better than prior multi-agent baselines using 4 GPUs. Through extensive evaluations on diverse benchmarks (MMLU-Pro, GPQA, AIME, and MedMCQA), we demonstrate that Symbolic-MoE outperforms strong LLMs like GPT4o-mini, as well as multi-agent approaches, with an absolute average improvement of 8.15% over the best multi-agent baseline. Moreover, Symbolic-MoE removes the need for expensive multi-round discussions, outperforming discussion baselines with less computation.
Large reasoning models (LRMs) like OpenAI-o1 and DeepSeek-R1 have demonstrated remarkable capabilities in complex reasoning tasks through the utilization of long Chain-of-thought (CoT). However, these models often suffer from hallucinations and inefficiencies due to their reliance solely on internal reasoning processes. In this paper, we introduce START (Self-Taught Reasoner with Tools), a novel tool-integrated long CoT reasoning LLM that significantly enhances reasoning capabilities by leveraging external tools. Through code execution, START is capable of performing complex computations, self-checking, exploring diverse methods, and self-debugging, thereby addressing the limitations of LRMs. The core innovation of START lies in its self-learning framework, which comprises two key techniques: 1) Hint-infer: We demonstrate that inserting artificially designed hints (e.g., ``Wait, maybe using Python here is a good idea.'') during the inference process of a LRM effectively stimulates its ability to utilize external tools without the need for any demonstration data. Hint-infer can also serve as a simple and effective sequential test-time scaling method; 2) Hint Rejection Sampling Fine-Tuning (Hint-RFT): Hint-RFT combines Hint-infer and RFT by scoring, filtering, and modifying the reasoning trajectories with tool invocation generated by a LRM via Hint-infer, followed by fine-tuning the LRM. Through this framework, we have fine-tuned the QwQ-32B model to achieve START. On PhD-level science QA (GPQA), competition-level math benchmarks (AMC23, AIME24, AIME25), and the competition-level code benchmark (LiveCodeBench), START achieves accuracy rates of 63.6%, 95.0%, 66.7%, 47.1%, and 47.3%, respectively. It significantly outperforms the base QwQ-32B and achieves performance comparable to the state-of-the-art open-weight model R1-Distill-Qwen-32B and the proprietary model o1-Preview.
The ability of large language models to solve complex mathematical problems has progressed significantly, particularly for tasks requiring advanced reasoning. However, the scarcity of sufficiently challenging problems, particularly at the Olympiad level, hinders further advancements. In this work, we introduce PromptCoT, a novel approach for automatically generating high-quality Olympiad-level math problems. The proposed method synthesizes complex problems based on mathematical concepts and the rationale behind problem construction, emulating the thought processes of experienced problem designers. We provide a theoretical analysis demonstrating that an optimal rationale should maximize both the likelihood of rationale generation given the associated concepts and the likelihood of problem generation conditioned on both the rationale and the concepts. Our method is evaluated on standard benchmarks including GSM8K, MATH-500, and AIME2024, where it consistently outperforms existing problem generation methods. Furthermore, we demonstrate that PromptCoT exhibits superior data scalability, consistently maintaining high performance as the dataset size increases, outperforming the baselines. The implementation is available at https://github.com/zhaoxlpku/PromptCoT.
Advances in large language models (LLMs) offer new possibilities for enhancing math education by automating support for both teachers and students. While prior work has focused on generating math problems and high-quality distractors, the role of visualization in math learning remains under-explored. Diagrams are essential for mathematical thinking and problem-solving, yet manually creating them is time-consuming and requires domain-specific expertise, limiting scalability. Recent research on using LLMs to generate Scalable Vector Graphics (SVG) presents a promising approach to automating diagram creation. Unlike pixel-based images, SVGs represent geometric figures using XML, allowing seamless scaling and adaptability. Educational platforms such as Khan Academy and IXL already use SVGs to display math problems and hints. In this paper, we explore the use of LLMs to generate math-related diagrams that accompany textual hints via intermediate SVG representations. We address three research questions: (1) how to automatically generate math diagrams in problem-solving hints and evaluate their quality, (2) whether SVG is an effective intermediate representation for math diagrams, and (3) what prompting strategies and formats are required for LLMs to generate accurate SVG-based diagrams. Our contributions include defining the task of automatically generating SVG-based diagrams for math hints, developing an LLM prompting-based pipeline, and identifying key strategies for improving diagram generation. Additionally, we introduce a Visual Question Answering-based evaluation setup and conduct ablation studies to assess different pipeline variations. By automating the math diagram creation, we aim to provide students and teachers with accurate, conceptually relevant visual aids that enhance problem-solving and learning experiences.
Training models to effectively use test-time compute is crucial for improving the reasoning performance of LLMs. Current methods mostly do so via fine-tuning on search traces or running RL with 0/1 outcome reward, but do these approaches efficiently utilize test-time compute? Would these approaches continue to scale as the budget improves? In this paper, we try to answer these questions. We formalize the problem of optimizing test-time compute as a meta-reinforcement learning (RL) problem, which provides a principled perspective on spending test-time compute. This perspective enables us to view the long output stream from the LLM as consisting of several episodes run at test time and leads us to use a notion of cumulative regret over output tokens as a way to measure the efficacy of test-time compute. Akin to how RL algorithms can best tradeoff exploration and exploitation over training, minimizing cumulative regret would also provide the best balance between exploration and exploitation in the token stream. While we show that state-of-the-art models do not minimize regret, one can do so by maximizing a dense reward bonus in conjunction with the outcome 0/1 reward RL. This bonus is the ''progress'' made by each subsequent block in the output stream, quantified by the change in the likelihood of eventual success. Using these insights, we develop Meta Reinforcement Fine-Tuning, or MRT, a new class of fine-tuning methods for optimizing test-time compute. MRT leads to a 2-3x relative gain in performance and roughly a 1.5x gain in token efficiency for math reasoning compared to outcome-reward RL.
DeepSeek-R1-Zero has successfully demonstrated the emergence of reasoning capabilities in LLMs purely through Reinforcement Learning (RL). Inspired by this breakthrough, we explore how RL can be utilized to enhance the reasoning capability of MLLMs. However, direct training with RL struggles to activate complex reasoning capabilities such as questioning and reflection in MLLMs, due to the absence of substantial high-quality multimodal reasoning data. To address this issue, we propose the reasoning MLLM, Vision-R1, to improve multimodal reasoning capability. Specifically, we first construct a high-quality multimodal CoT dataset without human annotations by leveraging an existing MLLM and DeepSeek-R1 through modality bridging and data filtering to obtain a 200K multimodal CoT dataset, Vision-R1-cold dataset. It serves as cold-start initialization data for Vision-R1. To mitigate the optimization challenges caused by overthinking after cold start, we propose Progressive Thinking Suppression Training (PTST) strategy and employ Group Relative Policy Optimization (GRPO) with the hard formatting result reward function to gradually refine the model's ability to learn correct and complex reasoning processes on a 10K multimodal math dataset. Comprehensive experiments show our model achieves an average improvement of $\sim$6% across various multimodal math reasoning benchmarks. Vision-R1-7B achieves a 73.5% accuracy on the widely used MathVista benchmark, which is only 0.4% lower than the leading reasoning model, OpenAI O1. The datasets and code will be released in: https://github.com/Osilly/Vision-R1 .
As specialized large language models (LLMs) become increasingly prevalent, model merging methods are being used to combine them to create a single multi-task model without requiring any additional data or training. However, these approaches fall short when the objective of merging is to increase the downstream model's performance on a particular task-specific benchmark. In this work, we propose LEWIS (Layer Wise Sparsity), a guided model-merging framework that uses activation-based layer importance to dynamically adjust layer-wise task-vector sparsity required for the merge process. LEWIS uses a calibration dataset to prioritize critical layers during the task-vector pruning process required for model merging. This approach guides existing merging methods by preserving essential layer-wise task-specific knowledge while ensuring the merged model performs the best at benchmarks resembling the calibration dataset. Our experiments demonstrate the effectiveness of LEWIS with performance improvements of code instruction-following and math-solving models created through model merging up to 4 percent and 11.3 percent, respectively, outperforming unguided data-less model merging approaches that use uniform-sparsity.
Advanced reasoning in large language models has achieved remarkable performance on challenging tasks, but the prevailing long-context reasoning paradigm faces critical limitations: quadratic computational scaling with sequence length, reasoning constrained by maximum context boundaries, and performance degradation beyond pre-training context windows. Existing approaches primarily compress reasoning chains without addressing the fundamental scaling problem. To overcome these challenges, we introduce InftyThink, a paradigm that transforms monolithic reasoning into an iterative process with intermediate summarization. By interleaving short reasoning segments with concise progress summaries, our approach enables unbounded reasoning depth while maintaining bounded computational costs. This creates a characteristic sawtooth memory pattern that significantly reduces computational complexity compared to traditional approaches. Furthermore, we develop a methodology for reconstructing long-context reasoning datasets into our iterative format, transforming OpenR1-Math into 333K training instances. Experiments across multiple model architectures demonstrate that our approach reduces computational costs while improving performance, with Qwen2.5-Math-7B showing 3-13% improvements across MATH500, AIME24, and GPQA_diamond benchmarks. Our work challenges the assumed trade-off between reasoning depth and computational efficiency, providing a more scalable approach to complex reasoning without architectural modifications.
This paper examines the intricate interplay among AI safety, security, and governance by integrating technical systems engineering with principles of moral imagination and ethical philosophy. Drawing on foundational insights from Weapons of Math Destruction and Thinking in Systems alongside contemporary debates in AI ethics, we develop a comprehensive multi-dimensional framework designed to regulate AI technologies deployed in high-stakes domains such as defense, finance, healthcare, and education. Our approach combines rigorous technical analysis, quantitative risk assessment, and normative evaluation to expose systemic vulnerabilities inherent in opaque, black-box models. Detailed case studies, including analyses of Microsoft Tay (2016) and the UK A-Level Grading Algorithm (2020), demonstrate how security lapses, bias amplification, and lack of accountability can precipitate cascading failures that undermine public trust. We conclude by outlining targeted strategies for enhancing AI resilience through adaptive regulatory mechanisms, robust security protocols, and interdisciplinary oversight, thereby advancing the state of the art in ethical and technical AI governance.
Large Language Models (LLMs) have displayed astonishing abilities in various tasks, especially in text generation, classification, question answering, etc. However, the reasoning ability of LLMs still faces many debates. The inherent ambiguity of Natural Language (NL) limits LLMs' ability to perform verifiable reasoning, making its answers lack coherence and trustworthy support. To tackle the above problems, we propose a novel framework named FANS: Formal ANswer Selection for Natural Language Math Reasoning Using Lean4. To the best of our knowledge, it is the first framework that utilizes Lean4 to enhance LLMs' NL math reasoning ability. In particular, given an NL math question and LLM-generated answers, FANS first translates it into Lean4 theorem statements. Then it tries to prove it using a Lean4 prover and verify it by Lean4. Finally, it uses the FL result to assist in answer selection. It enhances LLMs' NL math ability in providing a computer-verifiable solution for its correct answer and proposes an alternative method for answer selection beyond the reward model. Extensive experiments indicate the effectiveness of our framework. It can improve the accuracy rate of reward model enhanced LLMs in the MATH-500 dataset by at most 1.91% and AMC-23 by at most 8.33% on strong reward-model baselines. In some particular fields like number theory that Lean4 experts in, we can even select all correct solutions. The qualitative analysis also shows our framework can make NL results formally backed by Lean4 proofs. As a pioneering work in the corresponding field, we will open-source all our models and datasets to further boost the development of the field.
Reinforcement learning (RL) is pivotal for enabling large language models (LLMs) to generate long chains of thought (CoT) for complex tasks like math and reasoning. However, Proximal Policy Optimization (PPO), effective in many RL scenarios, fails in long CoT tasks. This paper identifies that value initialization bias and reward signal decay are the root causes of PPO's failure. We propose Value-Calibrated PPO (VC-PPO) to address these issues. In VC-PPO, the value model is pretrained to tackle initialization bias, and the Generalized Advantage Estimation (GAE) computation is decoupled between the actor and critic to mitigate reward signal decay. Experiments on the American Invitational Mathematics Examination (AIME) show that VC-PPO significantly boosts PPO performance. Ablation studies show that techniques in VC-PPO are essential in enhancing PPO for long CoT tasks.
We investigate the robustness of reasoning models trained for step-by-step problem solving by introducing query-agnostic adversarial triggers - short, irrelevant text that, when appended to math problems, systematically mislead models to output incorrect answers without altering the problem's semantics. We propose CatAttack, an automated iterative attack pipeline for generating triggers on a weaker, less expensive proxy model (DeepSeek V3) and successfully transfer them to more advanced reasoning target models like DeepSeek R1 and DeepSeek R1-distilled-Qwen-32B, resulting in greater than 300% increase in the likelihood of the target model generating an incorrect answer. For example, appending, "Interesting fact: cats sleep most of their lives," to any math problem leads to more than doubling the chances of a model getting the answer wrong. Our findings highlight critical vulnerabilities in reasoning models, revealing that even state-of-the-art models remain susceptible to subtle adversarial inputs, raising security and reliability concerns. The CatAttack triggers dataset with model responses is available at https://huggingface.co/datasets/collinear-ai/cat-attack-adversarial-triggers.
Uncertainty estimation is crucial for evaluating Large Language Models (LLMs), particularly in high-stakes domains where incorrect answers result in significant consequences. Numerous approaches consider this problem, while focusing on a specific type of uncertainty, ignoring others. We investigate what estimates, specifically token-wise entropy and model-as-judge (MASJ), would work for multiple-choice question-answering tasks for different question topics. Our experiments consider three LLMs: Phi-4, Mistral, and Qwen of different sizes from 1.5B to 72B and $14$ topics. While MASJ performs similarly to a random error predictor, the response entropy predicts model error in knowledge-dependent domains and serves as an effective indicator of question difficulty: for biology ROC AUC is $0.73$. This correlation vanishes for the reasoning-dependent domain: for math questions ROC-AUC is $0.55$. More principally, we found out that the entropy measure required a reasoning amount. Thus, data-uncertainty related entropy should be integrated within uncertainty estimates frameworks, while MASJ requires refinement. Moreover, existing MMLU-Pro samples are biased, and should balance required amount of reasoning for different subdomains to provide a more fair assessment of LLMs performance.
Multimodal Large Language Models (MLLMs) have shown promising capabilities in mathematical reasoning within visual contexts across various datasets. However, most existing multimodal math benchmarks are limited to single-visual contexts, which diverges from the multi-visual scenarios commonly encountered in real-world mathematical applications. To address this gap, we introduce MV-MATH: a meticulously curated dataset of 2,009 high-quality mathematical problems. Each problem integrates multiple images interleaved with text, derived from authentic K-12 scenarios, and enriched with detailed annotations. MV-MATH includes multiple-choice, free-form, and multi-step questions, covering 11 subject areas across 3 difficulty levels, and serves as a comprehensive and rigorous benchmark for assessing MLLMs' mathematical reasoning in multi-visual contexts. Through extensive experimentation, we observe that MLLMs encounter substantial challenges in multi-visual math tasks, with a considerable performance gap relative to human capabilities on MV-MATH. Furthermore, we analyze the performance and error patterns of various models, providing insights into MLLMs' mathematical reasoning capabilities within multi-visual settings.
We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle harder problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with support for non-constructive problems, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads (IMO) 2000-2024 geometry problems from 66% to 88%. The search process of AlphaGeometry2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that enables effective communication between search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AlphaGeometry2 to 84% for $\textit{all}$ geometry problems over the last 25 years, compared to 54% previously. AlphaGeometry2 was also part of the system that achieved silver-medal standard at IMO 2024 https://dpmd.ai/imo-silver. Last but not least, we report progress towards using AlphaGeometry2 as a part of a fully automated system that reliably solves geometry problems directly from natural language input.
Mathematical formulas are a fundamental and widely used component in various scientific fields, serving as a universal language for expressing complex concepts and relationships. While state-of-the-art transformer models excel in processing and understanding natural language, they encounter challenges with mathematical notation, which involves a complex structure and diverse representations. This study focuses on the development of specialized training datasets to enhance the encoding of mathematical content. We introduce Math Mutator (MAMUT), a framework capable of generating equivalent and falsified versions of a given mathematical formula in LaTeX notation, effectively capturing the mathematical variety in notation of the same concept. Based on MAMUT, we have generated four large mathematical datasets containing diverse notation, which can be used to train language models with enhanced mathematical embeddings.
This paper identifies the misinterpretation of the context can be a significant issue during the reasoning process of large language models, spanning from smaller models like Llama3.2-3B-Instruct to cutting-edge ones like DeepSeek-R1. For example, in the phrase "10 dollars per kilo," LLMs might not recognize that "per" means "for each," leading to calculation errors. We introduce a novel, post-training approach called **Stick to the Facts (SIFT)** to tackle this. SIFT leverages increasing inference-time compute to ground LLM reasoning in contexts. At the core of SIFT lies the *Sticker*, which is generated by the model itself to explicitly emphasize the key information within the context. Given the curated Sticker, SIFT generates two predictions -- one from the original query and one from the query augmented with the Sticker. If they differ, the Sticker is sequentially refined via *forward* optimization (to better align the extracted facts with the query) and *inverse* generation (to conform with the model's inherent tendencies) for more faithful reasoning outcomes. Studies across diverse models (from 3B to 100B+) and benchmarks (e.g., GSM8K, MATH-500) reveal consistent performance improvements. Notably, SIFT improves the pass@1 accuracy of DeepSeek-R1 on AIME2024 from 78.33% to **85.67**%, establishing a new state-of-the-art in the open-source community. The code is available at https://github.com/zhijie-group/SIFT.