Math Word Problem (MWP) solving is a critical task in natural language processing, has garnered significant research interest in recent years. Various recent studies heavily rely on Seq2Seq models and their extensions (e.g., Seq2Tree and Graph2Tree) to generate mathematical equations. While effective, these models struggle to generate diverse but counterpart solution equations, limiting their generalization across various math problem scenarios. In this paper, we introduce a novel Diversity-enhanced Knowledge Distillation (DivKD) model for practical MWP solving. Our approach proposes an adaptive diversity distillation method, in which a student model learns diverse equations by selectively transferring high-quality knowledge from a teacher model. Additionally, we design a diversity prior-enhanced student model to better capture the diversity distribution of equations by incorporating a conditional variational auto-encoder. Extensive experiments on {four} MWP benchmark datasets demonstrate that our approach achieves higher answer accuracy than strong baselines while maintaining high efficiency for practical applications.
Many students struggle with math word problems (MWPs), often finding it difficult to identify key information and select the appropriate mathematical operations. Schema-based instruction (SBI) is an evidence-based strategy that helps students categorize problems based on their structure, improving problem-solving accuracy. Building on this, we propose a Schema-Based Instruction Retrieval-Augmented Generation (SBI-RAG) framework that incorporates a large language model (LLM). Our approach emphasizes step-by-step reasoning by leveraging schemas to guide solution generation. We evaluate its performance on the GSM8K dataset, comparing it with GPT-4 and GPT-3.5 Turbo, and introduce a "reasoning score" metric to assess solution quality. Our findings suggest that SBI-RAG enhances reasoning clarity and facilitates a more structured problem-solving process potentially providing educational benefits for students.
Large Language Models (LLMs) exhibit impressive performance across various domains but still struggle with arithmetic reasoning tasks. Recent work shows the effectiveness of prompt design methods in enhancing reasoning capabilities. However, these approaches overlook crucial requirements for prior knowledge of specific concepts, theorems, and tricks to tackle most arithmetic reasoning problems successfully. To address this issue, we propose a novel and effective Teaching-Inspired Integrated Framework, which emulates the instructional process of a teacher guiding students. This method equips LLMs with essential concepts, relevant theorems, and similar problems with analogous solution approaches, facilitating the enhancement of reasoning abilities. Additionally, we introduce two new Chinese datasets, MathMC and MathToF, both with detailed explanations and answers. Experiments are conducted on nine benchmarks which demonstrates that our approach improves the reasoning accuracy of LLMs. With GPT-4 and our framework, we achieve new state-of-the-art performance on four math benchmarks (AddSub, SVAMP, Math23K and AQuA) with accuracies of 98.2% (+3.3%), 93.9% (+0.2%), 94.3% (+7.2%) and 81.1% (+1.2%). Our data and code are available at https://github.com/SallyTan13/Teaching-Inspired-Prompting.
Solving mathematical problems requires advanced reasoning abilities and presents notable challenges for large language models. Previous works usually synthesize data from proprietary models to augment existing datasets, followed by instruction tuning to achieve top-tier results. However, our analysis of these datasets reveals severe biases towards easy queries, with frequent failures to generate any correct response for the most challenging queries. Hypothesizing that difficult queries are crucial to learn complex reasoning, we propose Difficulty-Aware Rejection Tuning (DART), a method that allocates difficult queries more trials during the synthesis phase, enabling more extensive training on difficult samples. Utilizing DART, we have created new datasets for mathematical problem-solving that focus more on difficult queries and are substantially smaller than previous ones. Remarkably, our synthesis process solely relies on a 7B-sized open-weight model, without reliance on the commonly used proprietary GPT-4. We fine-tune various base models on our datasets ranging from 7B to 70B in size, resulting in a series of strong models called DART-MATH. In comprehensive in-domain and out-of-domain evaluation on 6 mathematical benchmarks, DART-MATH outperforms vanilla rejection tuning significantly, being superior or comparable to previous arts, despite using much smaller datasets and no proprietary models. Furthermore, our results position our synthetic datasets as the most effective and cost-efficient publicly available resources for advancing mathematical problem-solving.
This report introduces the Qwen2 series, the latest addition to our large language models and large multimodal models. We release a comprehensive suite of foundational and instruction-tuned language models, encompassing a parameter range from 0.5 to 72 billion, featuring dense models and a Mixture-of-Experts model. Qwen2 surpasses most prior open-weight models, including its predecessor Qwen1.5, and exhibits competitive performance relative to proprietary models across diverse benchmarks on language understanding, generation, multilingual proficiency, coding, mathematics, and reasoning. The flagship model, Qwen2-72B, showcases remarkable performance: 84.2 on MMLU, 37.9 on GPQA, 64.6 on HumanEval, 89.5 on GSM8K, and 82.4 on BBH as a base language model. The instruction-tuned variant, Qwen2-72B-Instruct, attains 9.1 on MT-Bench, 48.1 on Arena-Hard, and 35.7 on LiveCodeBench. Moreover, Qwen2 demonstrates robust multilingual capabilities, proficient in approximately 30 languages, spanning English, Chinese, Spanish, French, German, Arabic, Russian, Korean, Japanese, Thai, Vietnamese, and more, underscoring its versatility and global reach. To foster community innovation and accessibility, we have made the Qwen2 model weights openly available on Hugging Face and ModelScope, and the supplementary materials including example code on GitHub. These platforms also include resources for quantization, fine-tuning, and deployment, facilitating a wide range of applications and research endeavors.
Mathematical reasoning presents a significant challenge for Large Language Models (LLMs) due to the extensive and precise chain of reasoning required for accuracy. Ensuring the correctness of each reasoning step is critical. To address this, we aim to enhance the robustness and factuality of LLMs by learning from human feedback. However, Direct Preference Optimization (DPO) has shown limited benefits for long-chain mathematical reasoning, as models employing DPO struggle to identify detailed errors in incorrect answers. This limitation stems from a lack of fine-grained process supervision. We propose a simple, effective, and data-efficient method called Step-DPO, which treats individual reasoning steps as units for preference optimization rather than evaluating answers holistically. Additionally, we have developed a data construction pipeline for Step-DPO, enabling the creation of a high-quality dataset containing 10K step-wise preference pairs. We also observe that in DPO, self-generated data is more effective than data generated by humans or GPT-4, due to the latter's out-of-distribution nature. Our findings demonstrate that as few as 10K preference data pairs and fewer than 500 Step-DPO training steps can yield a nearly 3% gain in accuracy on MATH for models with over 70B parameters. Notably, Step-DPO, when applied to Qwen2-72B-Instruct, achieves scores of 70.8% and 94.0% on the test sets of MATH and GSM8K, respectively, surpassing a series of closed-source models, including GPT-4-1106, Claude-3-Opus, and Gemini-1.5-Pro. Our code, data, and models are available at https://github.com/dvlab-research/Step-DPO.
Although recent advancements in large language models (LLMs) have significantly improved their performance on various tasks, they still face challenges with complex and symbolic multi-step reasoning, particularly in mathematical reasoning. To bolster the mathematical reasoning capabilities of LLMs, most existing efforts concentrate on seeking assistance from either domain experts or GPT-4 for high-quality process-supervised data, which is not only expensive but also labor-intensive. In our study, we propose an innovative framework, AlphaMath, that bypasses the need for process annotations (from humans or GPTs) by leveraging Monte Carlo Tree Search (MCTS). This framework focuses on unleashing the potential of a well-pretrained LLM to autonomously enhance its mathematical reasoning. Specifically, we integrate a value model with the LLM, automatically generating both process supervision and step-level evaluation signals in MCTS. Furthermore, we propose an efficient inference strategy, step-level beam search, where the value model is crafted to assist the policy model (i.e., LLM) in navigating more effective reasoning paths, rather than solely relying on prior probabilities. The experimental results on both in-domain and out-of-domain datasets demonstrate that even without GPT-4 or human-annotated process supervision, our AlphaMath framework achieves comparable or superior results to previous state-of-the-art methods.
Chain-of-Thought (CoT) prompting has enhanced the performance of Large Language Models (LLMs) across various reasoning tasks. However, CoT still falls short in dealing with complex math word problems, as it usually suffers from three pitfalls: semantic misunderstanding errors, calculation errors, and step-missing errors. Prior studies involve addressing the calculation errors and step-missing errors, but neglect the semantic misunderstanding errors, which is the major factor limiting the reasoning performance of LLMs. To this end, we propose a simple-yet-effective method, namely Deeply Understanding the Problems (DUP), to improve the LLMs' math problem-solving ability by addressing semantic misunderstanding errors. The core of our method is to encourage the LLMs to deeply understand the problems and extract the key problem-solving information used for better reasoning. Extensive experiments on 10 diverse reasoning benchmarks show that our DUP method consistently outperforms the other counterparts by a large margin. More encouragingly, DUP achieves a new SOTA result on the GSM8K benchmark, with an accuracy of 97.1% under the zero-shot setting.
Despite the impressive capabilities of Large Language Models (LLMs) on various tasks, they still struggle with scenarios that involves complex reasoning and planning. Recent work proposed advanced prompting techniques and the necessity of fine-tuning with high-quality data to augment LLMs' reasoning abilities. However, these approaches are inherently constrained by data availability and quality. In light of this, self-correction and self-learning emerge as viable solutions, employing strategies that allow LLMs to refine their outputs and learn from self-assessed rewards. Yet, the efficacy of LLMs in self-refining its response, particularly in complex reasoning and planning task, remains dubious. In this paper, we introduce AlphaLLM for the self-improvements of LLMs, which integrates Monte Carlo Tree Search (MCTS) with LLMs to establish a self-improving loop, thereby enhancing the capabilities of LLMs without additional annotations. Drawing inspiration from the success of AlphaGo, AlphaLLM addresses the unique challenges of combining MCTS with LLM for self-improvement, including data scarcity, the vastness search spaces of language tasks, and the subjective nature of feedback in language tasks. AlphaLLM is comprised of prompt synthesis component, an efficient MCTS approach tailored for language tasks, and a trio of critic models for precise feedback. Our experimental results in mathematical reasoning tasks demonstrate that AlphaLLM significantly enhances the performance of LLMs without additional annotations, showing the potential for self-improvement in LLMs.
Recent advancements in large language models, such as GPT-4, have demonstrated remarkable capabilities in processing standard queries. Despite these advancements, their performance substantially declines in \textbf{advanced mathematical problems requiring complex, multi-step logical reasoning}. To enhance their inferential capabilities, current research has delved into \textit{prompting engineering}, exemplified by methodologies such as the Tree of Thought and Graph of Thought. Nonetheless, these existing approaches encounter two significant limitations. Firstly, their effectiveness in tackling complex mathematical problems is somewhat constrained. Secondly, the necessity to design distinct prompts for individual problems hampers their generalizability. In response to these limitations, this paper introduces the \textit{Multi-Agent System for conditional Mining} (\textbf{MACM}) prompting method. It not only resolves intricate mathematical problems but also demonstrates strong generalization capabilities across various mathematical contexts. With the assistance of MACM, the accuracy of GPT-4 Turbo on the most challenging level five mathematical problems in the MATH dataset increase from $\mathbf{54.68\%} \text{ to } \mathbf{76.73\%}$. The code is available in \url{https://github.com/bin123apple/MACM}.
Large language models (LLMs) are displaying emergent abilities for math reasoning tasks,and there is a growing attention on enhancing the ability of open-source LLMs through supervised fine-tuning (SFT).In this paper, we aim to explore a general data strategy for supervised data to help optimize and expand math reasoning ability.Firstly, we determine the ability boundary of reasoning paths augmentation by identifying these paths' minimal optimal set.Secondly, we validate that different abilities of the model can be cumulatively enhanced by Mix of Minimal Optimal Sets of corresponding types of data, while our models MMOS achieve SOTA performance on series base models under much lower construction costs.Besides, we point out GSM-HARD is not really hard and today's LLMs no longer lack numerical robustness.Also, we provide an Auto Problem Generator for robustness testing and educational applications.Our code and data are publicly available at https://github.com/cyzhh/MMOS.
Large language models (LLMs) have achieved impressive performance across various mathematical reasoning benchmarks. However, there are increasing debates regarding whether these models truly understand and apply mathematical knowledge or merely rely on shortcuts for mathematical reasoning. One essential and frequently occurring evidence is that when the math questions are slightly changed, LLMs can behave incorrectly. This motivates us to evaluate the robustness of LLMs' math reasoning capability by testing a wide range of question variations. We introduce the adversarial grade school math (GSM-Plus) dataset, an extension of GSM8K augmented with various mathematical perturbations. Our experiments on 25 LLMs and 4 prompting techniques show that while LLMs exhibit different levels of math reasoning abilities, their performances are far from robust. In particular, even for problems that have been solved in GSM8K, LLMs can make mistakes when new statements are added or the question targets are altered. We also explore whether more robust performance can be achieved by composing existing prompting methods, in which we try an iterative method that generates and verifies each intermediate thought based on its reasoning goal and calculation result.
Recent work has shown the immense potential of synthetically generated datasets for training large language models (LLMs), especially for acquiring targeted skills. Current large-scale math instruction tuning datasets such as MetaMathQA (Yu et al., 2024) and MAmmoTH (Yue et al., 2024) are constructed using outputs from closed-source LLMs with commercially restrictive licenses. A key reason limiting the use of open-source LLMs in these data generation pipelines has been the wide gap between the mathematical skills of the best closed-source LLMs, such as GPT-4, and the best open-source LLMs. Building on the recent progress in open-source LLMs, our proposed prompting novelty, and some brute-force scaling, we construct OpenMathInstruct-1, a math instruction tuning dataset with 1.8M problem-solution pairs. The dataset is constructed by synthesizing code-interpreter solutions for GSM8K and MATH, two popular math reasoning benchmarks, using the recently released and permissively licensed Mixtral model. Our best model, OpenMath-CodeLlama-70B, trained on a subset of OpenMathInstruct-1, achieves a score of 84.6% on GSM8K and 50.7% on MATH, which is competitive with the best gpt-distilled models. We release our code, models, and the OpenMathInstruct-1 dataset under a commercially permissive license.
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.
Despite the advancements in large language models (LLMs) for mathematical reasoning, solving competition-level math problems remains a significant challenge, especially for open-source LLMs without external tools. We introduce the MMIQC dataset, comprising a mixture of processed web data and synthetic question-response pairs, aimed at enhancing the mathematical reasoning capabilities of base language models. Models fine-tuned on MMIQC consistently surpass their counterparts in performance on the MATH benchmark across various model sizes. Notably, Qwen-72B-MMIQC achieves a 45.0% accuracy, exceeding the previous open-source state-of-the-art by 8.2% and outperforming the initial version GPT-4 released in 2023. Extensive evaluation results on Hungarian high school finals suggest that such improvement can generalize to unseen data. Our ablation study on MMIQC reveals that a large part of the improvement can be attributed to our novel augmentation method, Iterative Question Composing (IQC), which involves iteratively composing new questions from seed problems using an LLM and applying rejection sampling through another LLM.
Large language models (LLMs) have demonstrated considerable proficiency in general natural language processing (NLP) tasks. Instruction tuning, a successful paradigm, enhances the ability of LLMs to follow natural language instructions and exhibit robust generalization across general tasks. However, these models often encounter performance limitations across multiple tasks due to constrained model capacity. Expanding this capacity during the instruction tuning phase poses significant challenges. To address this issue, we introduce parameter-efficient sparsity crafting (PESC), which crafts dense models into sparse models using the mixture-of-experts (MoE) architecture. PESC integrates adapters into the MoE layers of sparse models, differentiating experts without altering the individual weights within these layers. This method significantly reduces computational costs and GPU memory requirements, facilitating model capacity expansion through a minimal parameter increase when guaranteeing the quality of approximation in function space compared to original sparse upcycling. Our empirical evaluation demonstrates the effectiveness of the PESC method. Using PESC during instruction tuning, our best sparse model outperforms other sparse and dense models and exhibits superior general capabilities compared to GPT-3.5. Our code is available at https://github.com/wuhy68/Parameter-Efficient-MoE.
Large Language Models (LLM) exhibit zero-shot mathematical reasoning capacity as a behavior emergent with scale, commonly manifesting as chain-of-thoughts (CoT) reasoning. However, multiple empirical findings suggest that this prowess is exclusive to LLMs with exorbitant sizes (beyond 50 billion parameters). Meanwhile, educational neuroscientists suggest that symbolic algebraic manipulation be introduced around the same time as arithmetic word problems to modularize language-to-formulation, symbolic manipulation of the formulation, and endgame arithmetic. In this paper, we start with the hypothesis that much smaller LMs, which are weak at multi-step reasoning, can achieve reasonable arithmetic reasoning if arithmetic word problems are posed as a formalize-then-solve task. In our architecture, which we call SYRELM, the LM serves the role of a translator to map natural language arithmetic questions into a formal language (FL) description. A symbolic solver then evaluates the FL expression to obtain the answer. A small frozen LM, equipped with an efficient low-rank adapter, is capable of generating FL expressions that incorporate natural language descriptions of the arithmetic problem (e.g., variable names and their purposes, formal expressions combining variables, etc.). We adopt policy-gradient reinforcement learning to train the adapted LM, informed by the non-differentiable symbolic solver. This marks a sharp departure from the recent development in tool-augmented LLMs, in which the external tools (e.g., calculator, Web search, etc.) are essentially detached from the learning phase of the LM. SYRELM shows massive improvements (e.g., +30.65 absolute point improvement in accuracy on the SVAMP dataset using GPT-J 6B model) over base LMs, while keeping our testbed easy to diagnose, interpret and within reach of most researchers.
We introduce FinanceMath, a novel benchmark designed to evaluate LLMs' capabilities in solving knowledge-intensive math reasoning problems. Compared to prior works, this study features three core advancements. First, FinanceMath includes 1,200 problems with a hybrid of textual and tabular content. These problems require college-level knowledge in the finance domain for effective resolution. Second, we provide expert-annotated, detailed solution references in Python program format, ensuring a high-quality benchmark for LLM assessment. We also construct a finance-domain knowledge bank and investigate various knowledge integration strategies. Finally, we evaluate a wide spectrum of 44 LLMs with both Chain-of-Thought and Program-of-Thought prompting methods. Our experimental results reveal that the current best-performing system (i.e., GPT-4o) achieves only 60.9% accuracy using CoT prompting, leaving substantial room for improvement. Moreover, while augmenting LLMs with external knowledge can improve model performance (e.g., from 47.5% to 54.5% for Gemini-1.5-Pro), their accuracy remains significantly lower than the estimated human expert performance of 92%. We believe that FinanceMath can advance future research in the area of domain-specific knowledge retrieval and integration, particularly within the context of solving reasoning-intensive tasks.
Large Language Models (LLMs), combined with program-based solving techniques, are increasingly demonstrating proficiency in mathematical reasoning. For example, closed-source models such as OpenAI GPT-4 and Claude show excellent results in solving math word problems. However, progress in math word problem-solving for open-source LLMs is limited, and the challenges these models face are not well-studied. In this paper, we study the performance of strong open-source LLMs, including Llama 2 (7B), Code Llama (7B), and Mistral (7B) on math word problems using program-based solving techniques. Specifically, we analyze the outputs of these models when applied to math word problems and identify a category of problems that pose a significant challenge, particularly those involving quantities spanning multiple units. To address this issue, we propose a systematic approach by defining the units for each quantity and ensuring the consistency of these units during mathematical operations. We developed Unit Consistency Programs (UCPs), an annotated dataset of math word problems, each paired with programs containing unit specifications and unit verification routines. We fine-tuned Llama 2 (7B), Code Llama (7B), and Mistral (7B) models with UCPs to produce theirVerityMath variants. Our findings indicate that our approach, which incorporates unit consistency, currently slightly underperforms compared to an approach that does not. To understand the reasons behind this, we conduct an in-depth error analysis and suggest options for future improvements. Our code and dataset are available at https://github.com/vernontoh/VerityMath.
Solving math word problems depends on how to articulate the problems, the lens through which models view human linguistic expressions. Real-world settings count on such a method even more due to the diverse practices of the same mathematical operations. Earlier works constrain available thinking processes by limited prediction strategies without considering their significance in acquiring mathematical knowledge. We introduce Attention-based THought Expansion Network Architecture (ATHENA) to tackle the challenges of real-world practices by mimicking human thought expansion mechanisms in the form of neural network propagation. A thought expansion recurrently generates the candidates carrying the thoughts of possible math expressions driven from the previous step and yields reasonable thoughts by selecting the valid pathways to the goal. Our experiments show that ATHENA achieves a new state-of-the-art stage toward the ideal model that is compelling in variant questions even when the informativeness in training examples is restricted.