Autoformalization involves automatically translating informal math into formal theorems and proofs that are machine-verifiable. Euclidean geometry provides an interesting and controllable domain for studying autoformalization. In this paper, we introduce a neuro-symbolic framework for autoformalizing Euclidean geometry, which combines domain knowledge, SMT solvers, and large language models (LLMs). One challenge in Euclidean geometry is that informal proofs rely on diagrams, leaving gaps in texts that are hard to formalize. To address this issue, we use theorem provers to fill in such diagrammatic information automatically, so that the LLM only needs to autoformalize the explicit textual steps, making it easier for the model. We also provide automatic semantic evaluation for autoformalized theorem statements. We construct LeanEuclid, an autoformalization benchmark consisting of problems from Euclid's Elements and the UniGeo dataset formalized in the Lean proof assistant. Experiments with GPT-4 and GPT-4V show the capability and limitations of state-of-the-art LLMs on autoformalizing geometry problems. The data and code are available at https://github.com/loganrjmurphy/LeanEuclid.
The rapid development of large language and vision models (LLVMs) has been driven by advances in visual instruction tuning. Recently, open-source LLVMs have curated high-quality visual instruction tuning datasets and utilized additional vision encoders or multiple computer vision models in order to narrow the performance gap with powerful closed-source LLVMs. These advancements are attributed to multifaceted information required for diverse capabilities, including fundamental image understanding, real-world knowledge about common-sense and non-object concepts (e.g., charts, diagrams, symbols, signs, and math problems), and step-by-step procedures for solving complex questions. Drawing from the multifaceted information, we present a new efficient LLVM, Mamba-based traversal of rationales (Meteor), which leverages multifaceted rationale to enhance understanding and answering capabilities. To embed lengthy rationales containing abundant information, we employ the Mamba architecture, capable of processing sequential data with linear time complexity. We introduce a new concept of traversal of rationale that facilitates efficient embedding of rationale. Subsequently, the backbone multimodal language model (MLM) is trained to generate answers with the aid of rationale. Through these steps, Meteor achieves significant improvements in vision language performances across multiple evaluation benchmarks requiring diverse capabilities, without scaling up the model size or employing additional vision encoders and computer vision models.
Math Word Problems (MWPs) play a vital role in assessing the capabilities of Large Language Models (LLMs), yet current research primarily focuses on questions with concise contexts. The impact of longer contexts on mathematical reasoning remains under-explored. This study pioneers the investigation of Context Length Generalizability (CoLeG), which refers to the ability of LLMs to solve MWPs with extended narratives. We introduce Extended Grade-School Math (E-GSM), a collection of MWPs featuring lengthy narratives, and propose two novel metrics to evaluate the efficacy and resilience of LLMs in tackling these problems. Our analysis of existing zero-shot prompting techniques with proprietary LLMs along with open-source LLMs reveals a general deficiency in CoLeG. To alleviate these issues, we propose tailored approaches for different categories of LLMs. For proprietary LLMs, we introduce a new instructional prompt designed to mitigate the impact of long contexts. For open-source LLMs, we develop a novel auxiliary task for fine-tuning to enhance CoLeG. Our comprehensive results demonstrate the effectiveness of our proposed methods, showing improved performance on E-GSM. Additionally, we conduct an in-depth analysis to differentiate the effects of semantic understanding and reasoning efficacy, showing that our methods improves the latter. We also establish the generalizability of our methods across several other MWP benchmarks. Our findings highlight the limitations of current LLMs and offer practical solutions correspondingly, paving the way for further exploration of model generalizability and training methodologies.
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 \url{https://github.com/RUCAIBox/JiuZhang3.0}.
Recent advancements in large language models (LLMs) have showcased significant improvements in mathematics. However, traditional math benchmarks like GSM8k offer a unidimensional perspective, falling short in providing a holistic assessment of the LLMs' math capabilities. To address this gap, we introduce MathBench, a new benchmark that rigorously assesses the mathematical capabilities of large language models. MathBench spans a wide range of mathematical disciplines, offering a detailed evaluation of both theoretical understanding and practical problem-solving skills. The benchmark progresses through five distinct stages, from basic arithmetic to college mathematics, and is structured to evaluate models at various depths of knowledge. Each stage includes theoretical questions and application problems, allowing us to measure a model's mathematical proficiency and its ability to apply concepts in practical scenarios. MathBench aims to enhance the evaluation of LLMs' mathematical abilities, providing a nuanced view of their knowledge understanding levels and problem solving skills in a bilingual context. The project is released at https://github.com/open-compass/MathBench .
We present GSM-MC, a multiple-choice (MC) dataset constructed by collecting answers and incorrect predictions on GSM8K from 60 open-source models. Through extensive experiments, we show that LLMs' performance on the MC version of this popular benchmark is strongly correlated with their performance on the original version and is quite robust to distractor choices and option orders, while the evaluation time is reduced by a factor of up to 30. Following similar procedures, we introduce MATH-MC, constructed from MATH, and PythonIO, a new program reasoning MC dataset constructed from HumanEval and MBPP. Experimental results indicate that LLMs' performance on these MC benchmarks leaves much room for improvement. Our data and code are available at https://github.com/Geralt-Targaryen/MC-Evaluation.
The current safeguard mechanisms for large language models (LLMs) are indeed susceptible to jailbreak attacks, making them inherently fragile. Even the process of fine-tuning on apparently benign data for downstream tasks can jeopardize safety. One potential solution is to conduct safety fine-tuning subsequent to downstream fine-tuning. However, there's a risk of catastrophic forgetting during safety fine-tuning, where LLMs may regain safety measures but lose the task-specific knowledge acquired during downstream fine-tuning. In this paper, we introduce a safety realignment framework through subspace-oriented model fusion (SOMF), aiming to combine the safeguard capabilities of initially aligned model and the current fine-tuned model into a realigned model. Our approach begins by disentangling all task vectors from the weights of each fine-tuned model. We then identify safety-related regions within these vectors by subspace masking techniques. Finally, we explore the fusion of the initial safely aligned LLM with all task vectors based on the identified safety subspace. We validate that our safety realignment framework satisfies the safety requirements of a single fine-tuned model as well as multiple models during their fusion. Our findings confirm that SOMF preserves safety without notably compromising performance on downstream tasks, including instruction following in Chinese, English, and Hindi, as well as problem-solving capabilities in Code and Math.
Human cognition exhibits systematic compositionality, the algebraic ability to generate infinite novel combinations from finite learned components, which is the key to understanding and reasoning about complex logic. In this work, we investigate the compositionality of large language models (LLMs) in mathematical reasoning. Specifically, we construct a new dataset \textsc{MathTrap} by introducing carefully designed logical traps into the problem descriptions of MATH and GSM8K. Since problems with logical flaws are quite rare in the real world, these represent "unseen" cases to LLMs. Solving these requires the models to systematically compose (1) the mathematical knowledge involved in the original problems with (2) knowledge related to the introduced traps. Our experiments show that while LLMs possess both components of requisite knowledge, they do not \textbf{spontaneously} combine them to handle these novel cases. We explore several methods to mitigate this deficiency, such as natural language prompts, few-shot demonstrations, and fine-tuning. Additionally, we test the recently released OpenAI o1 model and find that human-like `slow thinking' helps improve the compositionality of LLMs. Overall, systematic compositionality remains an open challenge for large language models.
Language models, potentially augmented with tool usage such as retrieval are becoming the go-to means of answering questions. Understanding and answering questions in real-world settings often requires retrieving information from different sources, processing and aggregating data to extract insights, and presenting complex findings in form of structured artifacts such as novel tables, charts, or infographics. In this paper, we introduce TANQ, the first open domain question answering dataset where the answers require building tables from information across multiple sources. We release the full source attribution for every cell in the resulting table and benchmark state-of-the-art language models in open, oracle, and closed book setups. Our best-performing baseline, GPT4 reaches an overall F1 score of 29.1, lagging behind human performance by 19.7 points. We analyse baselines' performance across different dataset attributes such as different skills required for this task, including multi-hop reasoning, math operations, and unit conversions. We further discuss common failures in model-generated answers, suggesting that TANQ is a complex task with many challenges ahead.
Intelligent Tutoring Systems (ITSs) often contain an automated feedback component, which provides a predefined feedback message to students when they detect a predefined error. To such a feedback component, we often resort to template-based approaches. These approaches require significant effort from human experts to detect a limited number of possible student errors and provide corresponding feedback. This limitation is exemplified in open-ended math questions, where there can be a large number of different incorrect errors. In our work, we examine the capabilities of large language models (LLMs) to generate feedback for open-ended math questions, similar to that of an established ITS that uses a template-based approach. We fine-tune both open-source and proprietary LLMs on real student responses and corresponding ITS-provided feedback. We measure the quality of the generated feedback using text similarity metrics. We find that open-source and proprietary models both show promise in replicating the feedback they see during training, but do not generalize well to previously unseen student errors. These results suggest that despite being able to learn the formatting of feedback, LLMs are not able to fully understand mathematical errors made by students.
Geometry Problem Solving (GPS), which is a classic and challenging math problem, has attracted much attention in recent years. It requires a solver to comprehensively understand both text and diagram, master essential geometry knowledge, and appropriately apply it in reasoning. However, existing works follow a paradigm of neural machine translation and only focus on enhancing the capability of encoders, which neglects the essential characteristics of human geometry reasoning. In this paper, inspired by dual-process theory, we propose a Dual-Reasoning Geometry Solver (DualGeoSolver) to simulate the dual-reasoning process of humans for GPS. Specifically, we construct two systems in DualGeoSolver, namely Knowledge System and Inference System. Knowledge System controls an implicit reasoning process, which is responsible for providing diagram information and geometry knowledge according to a step-wise reasoning goal generated by Inference System. Inference System conducts an explicit reasoning process, which specifies the goal in each reasoning step and applies the knowledge to generate program tokens for resolving it. The two systems carry out the above process iteratively, which behaves more in line with human cognition. We conduct extensive experiments on two benchmark datasets, GeoQA and GeoQA+. The results demonstrate the superiority of DualGeoSolver in both solving accuracy and robustness from explicitly modeling human reasoning process and knowledge application.
Large Multimodal Models (LMMs) have achieved impressive success in visual understanding and reasoning, remarkably improving the performance of mathematical reasoning in a visual context. Yet, a challenging type of visual math lies in the multimodal graph theory problem, which demands that LMMs understand the graphical structures accurately and perform multi-step reasoning on the visual graph. Additionally, exploring multimodal graph theory problems will lead to more effective strategies in fields like biology, transportation, and robotics planning. To step forward in this direction, we are the first to design a benchmark named VisionGraph, used to explore the capabilities of advanced LMMs in solving multimodal graph theory problems. It encompasses eight complex graph problem tasks, from connectivity to shortest path problems. Subsequently, we present a Description-Program-Reasoning (DPR) chain to enhance the logical accuracy of reasoning processes through graphical structure description generation and algorithm-aware multi-step reasoning. Our extensive study shows that 1) GPT-4V outperforms Gemini Pro in multi-step graph reasoning; 2) All LMMs exhibit inferior perception accuracy for graphical structures, whether in zero/few-shot settings or with supervised fine-tuning (SFT), which further affects problem-solving performance; 3) DPR significantly improves the multi-step graph reasoning capabilities of LMMs and the GPT-4V (DPR) agent achieves SOTA performance.
Addressing the challenge of automated geometry math problem-solving in artificial intelligence (AI) involves understanding multi-modal information and mathematics. Current methods struggle with accurately interpreting geometry diagrams, which hinders effective problem-solving. To tackle this issue, we present the Geometry problem sOlver with natural Language Description (GOLD) model. GOLD enhances the extraction of geometric relations by separately processing symbols and geometric primitives within the diagram. Subsequently, it converts the extracted relations into natural language descriptions, efficiently utilizing large language models to solve geometry math problems. Experiments show that the GOLD model outperforms the Geoformer model, the previous best method on the UniGeo dataset, by achieving accuracy improvements of 12.7% and 42.1% in calculation and proving subsets. Additionally, it surpasses the former best model on the PGPS9K and Geometry3K datasets, PGPSNet, by obtaining accuracy enhancements of 1.8% and 3.2%, respectively.
We introduce an approach aimed at enhancing the reasoning capabilities of Large Language Models (LLMs) through an iterative preference learning process inspired by the successful strategy employed by AlphaZero. Our work leverages Monte Carlo Tree Search (MCTS) to iteratively collect preference data, utilizing its look-ahead ability to break down instance-level rewards into more granular step-level signals. To enhance consistency in intermediate steps, we combine outcome validation and stepwise self-evaluation, continually updating the quality assessment of newly generated data. The proposed algorithm employs Direct Preference Optimization (DPO) to update the LLM policy using this newly generated step-level preference data. Theoretical analysis reveals the importance of using on-policy sampled data for successful self-improving. Extensive evaluations on various arithmetic and commonsense reasoning tasks demonstrate remarkable performance improvements over existing models. For instance, our approach outperforms the Mistral-7B Supervised Fine-Tuning (SFT) baseline on GSM8K, MATH, and ARC-C, with substantial increases in accuracy to $81.8\%$ (+$5.9\%$), $34.7\%$ (+$5.8\%$), and $76.4\%$ (+$15.8\%$), respectively. Additionally, our research delves into the training and inference compute tradeoff, providing insights into how our method effectively maximizes performance gains. Our code is publicly available at https://github.com/YuxiXie/MCTS-DPO.
Recent advancements in large language models (LLMs) have showcased their exceptional abilities across various tasks, such as code generation, problem-solving and reasoning. Existing benchmarks evaluate tasks in isolation, yet the extent to which LLMs can understand prose-style tasks, identify the underlying problems, and then generate appropriate code solutions is still unexplored. Addressing this gap, we introduce PECC, a novel benchmark derived from Advent Of Code (AoC) challenges and Project Euler, including 2396 problems. Unlike conventional benchmarks, PECC requires LLMs to interpret narrative-embedded problems, extract requirements, and generate executable code. A key feature of our dataset is the complexity added by natural language prompting in chat-based evaluations, mirroring real-world instruction ambiguities. Results show varying model performance between narrative and neutral problems, with specific challenges in the Euler math-based subset with GPT-3.5-Turbo passing 50% of the AoC challenges and only 8% on the Euler problems. By probing the limits of LLMs' capabilities, our benchmark provides a framework to monitor and assess the subsequent progress of LLMs as a universal problem solver.
Self-correction has emerged as a promising solution to boost the reasoning performance of large language models (LLMs), where LLMs refine their solutions using self-generated critiques that pinpoint the errors. This work explores whether small (<= 13B) language models (LMs) have the ability of self-correction on reasoning tasks with minimal inputs from stronger LMs. We propose a novel pipeline that prompts smaller LMs to collect self-correction data that supports the training of self-refinement abilities. First, we leverage correct solutions to guide the model in critiquing their incorrect responses. Second, the generated critiques, after filtering, are used for supervised fine-tuning of the self-correcting reasoner through solution refinement. Our experimental results show improved self-correction abilities of two models on five datasets spanning math and commonsense reasoning, with notable performance gains when paired with a strong GPT-4-based verifier, though limitations are identified when using a weak self-verifier for determining when to correct.
Artificial Intelligence (AI) models have emerged as another important audience for programming languages alongside humans and machines, as we enter the era of large language models (LLMs). LLMs can now perform well in coding competitions and even write programs like developers to solve various tasks, including mathematical problems. However, the grammar and layout of current programs are designed to cater the needs of human developers -- with many grammar tokens and formatting tokens being used to make the code easier for humans to read. While this is helpful, such a design adds unnecessary computational work for LLMs, as each token they either use or produce consumes computational resources. To improve inference efficiency and reduce computational costs, we propose the concept of AI-oriented grammar. This aims to represent code in a way that better suits the working mechanism of AI models. Code written with AI-oriented grammar discards formats and uses a minimum number of tokens to convey code semantics effectively. To demonstrate the feasibility of this concept, we explore and implement the first AI-oriented grammar for Python, named SimPy. SimPy is crafted by revising the original Python grammar through a series of heuristic rules. Programs written in SimPy maintain identical AST structures to those in standard Python. This allows for not only execution via a modified AST parser, but also seamless transformation between programs written in Python and SimPy, enabling human developers and LLMs to use Python and SimPy, respectively, when they need to collaborate. In the experiments, compared with Python, SimPy enables a reduction in token usage by 13.5% and 10.4% for CodeLlama and GPT-4, respectively, when completing the same set of code-related tasks. Additionally, these models can maintain or even improve their performance when using SimPy instead of Python for these tasks.
We present LayerSkip, an end-to-end solution to speed-up inference of large language models (LLMs). First, during training we apply layer dropout, with low dropout rates for earlier layers and higher dropout rates for later layers, and an early exit loss where all transformer layers share the same exit. Second, during inference, we show that this training recipe increases the accuracy of early exit at earlier layers, without adding any auxiliary layers or modules to the model. Third, we present a novel self-speculative decoding solution where we exit at early layers and verify and correct with remaining layers of the model. Our proposed self-speculative decoding approach has less memory footprint than other speculative decoding approaches and benefits from shared compute and activations of the draft and verification stages. We run experiments on different Llama model sizes on different types of training: pretraining from scratch, continual pretraining, finetuning on specific data domain, and finetuning on specific task. We implement our inference solution and show speedups of up to 2.16x on summarization for CNN/DM documents, 1.82x on coding, and 2.0x on TOPv2 semantic parsing task. We open source our code and checkpoints at https://github.com/facebookresearch/LayerSkip.
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.
Large language models (LLMs) have been explored in a variety of reasoning tasks including solving of mathematical problems. Each math dataset typically includes its own specially designed evaluation script, which, while suitable for its intended use, lacks generalizability across different datasets. Consequently, updates and adaptations to these evaluation tools tend to occur without being systematically reported, leading to inconsistencies and obstacles to fair comparison across studies. To bridge this gap, we introduce a comprehensive mathematical evaluation toolkit that not only utilizes a python computer algebra system (CAS) for its numerical accuracy, but also integrates an optional LLM, known for its considerable natural language processing capabilities. To validate the effectiveness of our toolkit, we manually annotated two distinct datasets. Our experiments demonstrate that the toolkit yields more robust evaluation results compared to prior works, even without an LLM. Furthermore, when an LLM is incorporated, there is a notable enhancement. The code for our method will be made available at \url{https://github.com/MARIO-Math-Reasoning/math_evaluation}.