Large language models have consistently struggled with complex reasoning tasks, such as mathematical problem-solving. Investigating the internal reasoning mechanisms of these models can help us design better model architectures and training strategies, ultimately enhancing their reasoning capability. In this study, we constructed a symbolic dataset to investigate the mechanisms by which Transformer models employ vertical thinking strategy based on their inherent structure and horizontal thinking strategy based on Chain of Thought to achieve multi-step reasoning. We introduced the concept of buffer mechanism: the model stores various information in distinct buffers and selectively extracts them through the query-key matrix. We proposed a random matrix-based algorithm to enhance the model's reasoning ability, resulting in a 75% reduction in the training time required for the GPT-2 model to achieve generalization capability on the PrOntoQA dataset. These findings provide new insights into understanding the mechanisms of large language models.
Metacognitive knowledge refers to humans' intuitive knowledge of their own thinking and reasoning processes. Today's best LLMs clearly possess some reasoning processes. The paper gives evidence that they also have metacognitive knowledge, including ability to name skills and procedures to apply given a task. We explore this primarily in context of math reasoning, developing a prompt-guided interaction procedure to get a powerful LLM to assign sensible skill labels to math questions, followed by having it perform semantic clustering to obtain coarser families of skill labels. These coarse skill labels look interpretable to humans. To validate that these skill labels are meaningful and relevant to the LLM's reasoning processes we perform the following experiments. (a) We ask GPT-4 to assign skill labels to training questions in math datasets GSM8K and MATH. (b) When using an LLM to solve the test questions, we present it with the full list of skill labels and ask it to identify the skill needed. Then it is presented with randomly selected exemplar solved questions associated with that skill label. This improves accuracy on GSM8k and MATH for several strong LLMs, including code-assisted models. The methodology presented is domain-agnostic, even though this article applies it to math problems.
With the proliferation of large language models (LLMs), the comprehensive alignment of such models across multiple tasks has emerged as a critical area of research. Existing alignment methodologies primarily address single task, such as multi-turn dialogue, coding, mathematical problem-solving, and tool usage. However, AI-driven products that leverage language models usually necessitate a fusion of these abilities to function effectively in real-world scenarios. Moreover, the considerable computational resources required for proper alignment of LLMs underscore the need for a more robust, efficient, and encompassing approach to multi-task alignment, ensuring improved generative performance. In response to these challenges, we introduce a novel technique termed Mixture-of-Instructions (MoI), which employs a strategy of instruction concatenation combined with diverse system prompts to boost the alignment efficiency of language models. We have also compiled a diverse set of seven benchmark datasets to rigorously evaluate the alignment efficacy of the MoI-enhanced language model. Our methodology was applied to the open-source Qwen-7B-chat model, culminating in the development of Qwen-SFT-MoI. This enhanced model demonstrates significant advancements in generative capabilities across coding, mathematics, and tool use tasks.
Large Language Models (LLMs) have demonstrated remarkable performance across a spectrum of tasks. Recently, Direct Preference Optimization (DPO) has emerged as an RL-free approach to optimize the policy model on human preferences. However, several limitations hinder the widespread adoption of this method. To address these shortcomings, various versions of DPO have been introduced. Yet, a comprehensive evaluation of these variants across diverse tasks is still lacking. In this study, we aim to bridge this gap by investigating the performance of alignment methods across three distinct scenarios: (1) keeping the Supervised Fine-Tuning (SFT) part, (2) skipping the SFT part, and (3) skipping the SFT part and utilizing an instruction-tuned model. Furthermore, we explore the impact of different training sizes on their performance. Our evaluation spans a range of tasks including dialogue systems, reasoning, mathematical problem-solving, question answering, truthfulness, and multi-task understanding, encompassing 13 benchmarks such as MT-Bench, Big Bench, and Open LLM Leaderboard. Key observations reveal that alignment methods achieve optimal performance with smaller training data subsets, exhibit limited effectiveness in reasoning tasks yet significantly impact mathematical problem-solving, and employing an instruction-tuned model notably influences truthfulness. We anticipate that our findings will catalyze further research aimed at developing more robust models to address alignment challenges.
The rapid progress in the field of natural language processing (NLP) systems and the expansion of large language models (LLMs) have opened up numerous opportunities in the field of education and instructional methods. These advancements offer the potential for tailored learning experiences and immediate feedback, all delivered through accessible and cost-effective services. One notable application area for this technological advancement is in the realm of solving mathematical problems. Mathematical problem-solving not only requires the ability to decipher complex problem statements but also the skill to perform precise arithmetic calculations at each step of the problem-solving process. However, the evaluation of the arithmetic capabilities of large language models remains an area that has received relatively little attention. In response, we introduce an extensive mathematics dataset called "MathQuest" sourced from the 11th and 12th standard Mathematics NCERT textbooks. This dataset encompasses mathematical challenges of varying complexity and covers a wide range of mathematical concepts. Utilizing this dataset, we conduct fine-tuning experiments with three prominent LLMs: LLaMA-2, WizardMath, and MAmmoTH. These fine-tuned models serve as benchmarks for evaluating their performance on our dataset. Our experiments reveal that among the three models, MAmmoTH-13B emerges as the most proficient, achieving the highest level of competence in solving the presented mathematical problems. Consequently, MAmmoTH-13B establishes itself as a robust and dependable benchmark for addressing NCERT mathematics problems.
Large language models (LLMs) have shown excellent mastering of human language, but still struggle in real-world applications that require mathematical problem-solving. While many strategies and datasets to enhance LLMs' mathematics are developed, it remains a challenge to simultaneously maintain and improve both language and mathematical capabilities in deployed LLM systems.In this work, we tailor the Self-Critique pipeline, which addresses the challenge in the feedback learning stage of LLM alignment. We first train a general Math-Critique model from the LLM itself to provide feedback signals. Then, we sequentially employ rejective fine-tuning and direct preference optimization over the LLM's own generations for data collection. Based on ChatGLM3-32B, we conduct a series of experiments on both academic and our newly created challenging dataset, MathUserEval. Results show that our pipeline significantly enhances the LLM's mathematical problem-solving while still improving its language ability, outperforming LLMs that could be two times larger. Related techniques have been deployed to ChatGLM\footnote{\url{https://chatglm.cn}}, an online serving LLM. Related evaluation dataset and scripts are released at \url{https://github.com/THUDM/ChatGLM-Math}.
Large Language Models (LLMs) have demonstrated exceptional capabilities in various natural language tasks, often achieving performances that surpass those of humans. Despite these advancements, the domain of mathematics presents a distinctive challenge, primarily due to its specialized structure and the precision it demands. In this study, we adopted a two-step approach for investigating the proficiency of LLMs in answering mathematical questions. First, we employ the most effective LLMs, as identified by their performance on math question-answer benchmarks, to generate answers to 78 questions from the Math Stack Exchange (MSE). Second, a case analysis is conducted on the LLM that showed the highest performance, focusing on the quality and accuracy of its answers through manual evaluation. We found that GPT-4 performs best (nDCG of 0.48 and P@10 of 0.37) amongst existing LLMs fine-tuned for answering mathematics questions and outperforms the current best approach on ArqMATH3 Task1, considering P@10. Our Case analysis indicates that while the GPT-4 can generate relevant responses in certain instances, it does not consistently answer all questions accurately. This paper explores the current limitations of LLMs in navigating complex mathematical problem-solving. Through case analysis, we shed light on the gaps in LLM capabilities within mathematics, thereby setting the stage for future research and advancements in AI-driven mathematical reasoning. We make our code and findings publicly available for research: \url{https://github.com/gipplab/LLM-Investig-MathStackExchange}
Prompt compression is an innovative method for efficiently condensing input prompts while preserving essential information. To facilitate quick-start services, user-friendly interfaces, and compatibility with common datasets and metrics, we present the Prompt Compression Toolkit (PCToolkit). This toolkit is a unified plug-and-play solution for compressing prompts in Large Language Models (LLMs), featuring cutting-edge prompt compressors, diverse datasets, and metrics for comprehensive performance evaluation. PCToolkit boasts a modular design, allowing for easy integration of new datasets and metrics through portable and user-friendly interfaces. In this paper, we outline the key components and functionalities of PCToolkit. We conducted evaluations of the compressors within PCToolkit across various natural language tasks, including reconstruction, summarization, mathematical problem-solving, question answering, few-shot learning, synthetic tasks, code completion, boolean expressions, multiple choice questions, and lies recognition.
Despite the effectiveness of data selection for large language models (LLMs) during pretraining and instruction fine-tuning phases, improving data efficiency in supervised fine-tuning (SFT) for specialized domains poses significant challenges due to the complexity of fine-tuning data. To bridge this gap, we introduce an effective and scalable data selection method for SFT, SmallToLarge (S2L), which leverages training trajectories from small models to guide the data selection for larger models. We demonstrate through extensive experiments that S2L significantly improves data efficiency in SFT for mathematical problem-solving, reducing the training data to just 11% of the original MathInstruct dataset (Yue et al., 2023) to match full dataset performance while outperforming state-of-the-art data selection algorithms by an average of 4.7% across 6 in- and out-domain evaluation datasets. Remarkably, selecting only 50K data for SFT, S2L achieves a 32.7% accuracy on the most challenging MATH (Hendrycks et al., 2021) benchmark, improving Phi-2 (Li et al., 2023b) by 16.6%. In clinical text summarization on the MIMIC-III dataset (Johnson et al., 2016), S2L again outperforms training on the full dataset using only 50% of the data. Notably, S2L can perform data selection using a reference model 40x smaller than the target model, proportionally reducing the cost of data selection.
Large language models (LLMs) have accomplished remarkable reasoning performance in various domains. However, in the domain of reasoning tasks, we discover a frailty: LLMs are surprisingly brittle to the ordering of the premises, despite the fact that such ordering does not alter the underlying task. In particular, we observe that LLMs achieve the best performance when the premise order aligns with the context required in intermediate reasoning steps. For example, in deductive reasoning tasks, presenting the premises in the same order as the ground truth proof in the prompt (as opposed to random ordering) drastically increases the model's accuracy. We first examine the effect of premise ordering on deductive reasoning on a variety of LLMs, and our evaluation shows that permuting the premise order can cause a performance drop of over 30%. In addition, we release the benchmark R-GSM, based on GSM8K, to examine the ordering effect for mathematical problem-solving, and we again observe a significant drop in accuracy, relative to the original GSM8K benchmark.
Mathematical reasoning serves as a cornerstone for assessing the fundamental cognitive capabilities of human intelligence. In recent times, there has been a notable surge in the development of Large Language Models (LLMs) geared towards the automated resolution of mathematical problems. However, the landscape of mathematical problem types is vast and varied, with LLM-oriented techniques undergoing evaluation across diverse datasets and settings. This diversity makes it challenging to discern the true advancements and obstacles within this burgeoning field. This survey endeavors to address four pivotal dimensions: i) a comprehensive exploration of the various mathematical problems and their corresponding datasets that have been investigated; ii) an examination of the spectrum of LLM-oriented techniques that have been proposed for mathematical problem-solving; iii) an overview of factors and concerns affecting LLMs in solving math; and iv) an elucidation of the persisting challenges within this domain. To the best of our knowledge, this survey stands as one of the first extensive examinations of the landscape of LLMs in the realm of mathematics, providing a holistic perspective on the current state, accomplishments, and future challenges in this rapidly evolving field.
Large language models (LLMs) have shown remarkable proficiency in human-level reasoning and generation capabilities, which encourages extensive research on their application in mathematical problem solving. However, current work has been largely focused on text-based mathematical problems, with limited investigation in problems involving geometric information. Addressing this gap, we aim to enable LLMs to solve geometric problems by understanding image input. We first analyze the limitations of current Multimodal Large Language Models (MLLMs) in this area: they struggle to accurately comprehending basic geometric elements and their relationships. To overcome these challenges, we take advantage of the unique characteristics of geometric problems (such as unique geometric logical form, and geometric scalability) and the capacity of the textual LLMs to build an enriched multimodal geometry dataset based on existing data. The augmented dataset, Geo170K, contains more than 170K geometric image-caption and question-answer pairs. Utilizing our constructed Geo170K dataset, we develop G-LLaVA, which demonstrates exceptional performance in solving geometric problems, significantly outperforming GPT-4-V on the MathVista benchmark with only 7B parameters.
Due to the remarkable language understanding and generation abilities of large language models (LLMs), their use in educational applications has been explored. However, little work has been done on investigating the pedagogical ability of LLMs in helping students to learn mathematics. In this position paper, we discuss the challenges associated with employing LLMs to enhance students' mathematical problem-solving skills by providing adaptive feedback. Apart from generating the wrong reasoning processes, LLMs can misinterpret the meaning of the question, and also exhibit difficulty in understanding the given questions' rationales when attempting to correct students' answers. Three research questions are formulated.
Large Language Models (LLMs) have driven substantial progress in artificial intelligence in recent years, exhibiting impressive capabilities across a wide range of tasks, including mathematical problem-solving. Inspired by the success of subgoal-based methods, we propose a novel framework called \textbf{SE}quential sub\textbf{G}oal \textbf{O}ptimization (SEGO) to enhance LLMs' ability to solve mathematical problems. By establishing a connection between the subgoal breakdown process and the probability of solving problems, SEGO aims to identify better subgoals with theoretical guarantees. Addressing the challenge of identifying suitable subgoals in a large solution space, our framework generates problem-specific subgoals and adjusts them according to carefully designed criteria. Incorporating these optimized subgoals into the policy model training leads to significant improvements in problem-solving performance. We validate SEGO's efficacy through experiments on two benchmarks, GSM8K and MATH, where our approach outperforms existing methods, highlighting the potential of SEGO in AI-driven mathematical problem-solving. Data and code associated with this paper will be available at https://github.com/zhaoxlpku/SEGO
Recent claims about the impressive abilities of large language models (LLMs) are often supported by evaluating publicly available benchmarks. Since LLMs train on wide swaths of the internet, this practice raises concerns of data contamination, i.e., evaluating on examples that are explicitly or implicitly included in the training data. Data contamination remains notoriously challenging to measure and mitigate, even with partial attempts like controlled experimentation of training data, canary strings, or embedding similarities. In this work, we conduct the first thorough longitudinal analysis of data contamination in LLMs by using the natural experiment of training cutoffs in GPT models to look at benchmarks released over time. Specifically, we consider two code/mathematical problem-solving datasets, Codeforces and Project Euler, and find statistically significant trends among LLM pass rate vs. GitHub popularity and release date that provide strong evidence of contamination. By open-sourcing our dataset, raw results, and evaluation framework, our work paves the way for rigorous analyses of data contamination in modern models. We conclude with a discussion of best practices and future steps for publicly releasing benchmarks in the age of LLMs that train on webscale data.
When applied to question answering and other text generation tasks, language models (LMs) may be queried generatively (by sampling answers from their output distribution) or discriminatively (by using them to score or rank a set of candidate outputs). These procedures sometimes yield very different predictions. How do we reconcile mutually incompatible scoring procedures to obtain coherent LM predictions? We introduce a new, a training-free, game-theoretic procedure for language model decoding. Our approach casts language model decoding as a regularized imperfect-information sequential signaling game - which we term the CONSENSUS GAME - in which a GENERATOR seeks to communicate an abstract correctness parameter using natural language sentences to a DISCRIMINATOR. We develop computational procedures for finding approximate equilibria of this game, resulting in a decoding algorithm we call EQUILIBRIUM-RANKING. Applied to a large number of tasks (including reading comprehension, commonsense reasoning, mathematical problem-solving, and dialog), EQUILIBRIUM-RANKING consistently, and sometimes substantially, improves performance over existing LM decoding procedures - on multiple benchmarks, we observe that applying EQUILIBRIUM-RANKING to LLaMA-7B outperforms the much larger LLaMA-65B and PaLM-540B models. These results highlight the promise of game-theoretic tools for addressing fundamental challenges of truthfulness and consistency in LMs.
Large language models have made significant progress in various language tasks, yet they still struggle with complex mathematics. In this paper, we propose ToRA a series of Tool-integrated Reasoning Agents designed to solve challenging mathematical problems by seamlessly integrating natural language reasoning with the utilization of external tools (e.g., computation libraries and symbolic solvers), thereby amalgamating the analytical prowess of language and the computational efficiency of tools. To train ToRA, we curate interactive tool-use trajectories on mathematical datasets, apply imitation learning on the annotations, and propose output space shaping to further refine models' reasoning behavior. As a result, ToRA models significantly outperform open-source models on 10 mathematical reasoning datasets across all scales with 13%-19% absolute improvements on average. Notably, ToRA-7B reaches 44.6% on the competition-level dataset MATH, surpassing the best open-source model WizardMath-70B by 22% absolute. ToRA-Code-34B is also the first open-source model that achieves an accuracy exceeding 50% on MATH, which significantly outperforms GPT-4's CoT result, and is competitive with GPT-4 solving problems with programs. Additionally, we conduct a comprehensive analysis of the benefits and remaining challenges of tool interaction for mathematical reasoning, providing valuable insights for future research.
In the rapidly evolving landscape of education, digital technologies have repeatedly disrupted traditional pedagogical methods. This paper explores the latest of these disruptions: the potential integration of large language models (LLMs) and chatbots into graduate engineering education. We begin by tracing historical and technological disruptions to provide context and then introduce key terms such as machine learning and deep learning and the underlying mechanisms of recent advancements, namely attention/transformer models and graphics processing units. The heart of our investigation lies in the application of an LLM-based chatbot in a graduate fluid mechanics course. We developed a question bank from the course material and assessed the chatbot's ability to provide accurate, insightful responses. The results are encouraging, demonstrating not only the bot's ability to effectively answer complex questions but also the potential advantages of chatbot usage in the classroom, such as the promotion of self-paced learning, the provision of instantaneous feedback, and the reduction of instructors' workload. The study also examines the transformative effect of intelligent prompting on enhancing the chatbot's performance. Furthermore, we demonstrate how powerful plugins like Wolfram Alpha for mathematical problem-solving and code interpretation can significantly extend the chatbot's capabilities, transforming it into a comprehensive educational tool. While acknowledging the challenges and ethical implications surrounding the use of such AI models in education, we advocate for a balanced approach. The use of LLMs and chatbots in graduate education can be greatly beneficial but requires ongoing evaluation and adaptation to ensure ethical and efficient use.
This paper aims to investigate the mathematical problem-solving capabilities of Chat Generative Pre-Trained Transformer (ChatGPT) in case of Bayesian reasoning. The study draws inspiration from Zhu & Gigerenzer's research in 2006, which posed the question: Can children reason the Bayesian way? In the pursuit of answering this question, a set of 10 Bayesian reasoning problems were presented. The results of their work revealed that children's ability to reason effectively using Bayesian principles is contingent upon a well-structured information representation. In this paper, we present the same set of 10 Bayesian reasoning problems to ChatGPT. Remarkably, the results demonstrate that ChatGPT provides the right solutions to all problems.
Although pre-trained language models~(PLMs) have recently advanced the research progress in mathematical reasoning, they are not specially designed as a capable multi-task solver, suffering from high cost for multi-task deployment (\eg a model copy for a task) and inferior performance on complex mathematical problems in practical applications. To address these issues, in this paper, we propose \textbf{JiuZhang~2.0}, a unified Chinese PLM specially for multi-task mathematical problem solving. Our idea is to maintain a moderate-sized model and employ the \emph{cross-task knowledge sharing} to improve the model capacity in a multi-task setting. Specially, we construct a Mixture-of-Experts~(MoE) architecture for modeling mathematical text, so as to capture the common mathematical knowledge across tasks. For optimizing the MoE architecture, we design \emph{multi-task continual pre-training} and \emph{multi-task fine-tuning} strategies for multi-task adaptation. These training strategies can effectively decompose the knowledge from the task data and establish the cross-task sharing via expert networks. In order to further improve the general capacity of solving different complex tasks, we leverage large language models~(LLMs) as complementary models to iteratively refine the generated solution by our PLM, via in-context learning. Extensive experiments have demonstrated the effectiveness of our model.