The rapid development of large language models (LLMs) has spurred extensive research into their domain-specific capabilities, particularly mathematical reasoning. However, most open-source LLMs focus solely on mathematical reasoning, neglecting the integration with visual injection, despite the fact that many mathematical tasks rely on visual inputs such as geometric diagrams, charts, and function plots. To fill this gap, we introduce \textbf{MultiMath-7B}, a multimodal large language model that bridges the gap between math and vision. \textbf{MultiMath-7B} is trained through a four-stage process, focusing on vision-language alignment, visual and math instruction-tuning, and process-supervised reinforcement learning. We also construct a novel, diverse and comprehensive multimodal mathematical dataset, \textbf{MultiMath-300K}, which spans K-12 levels with image captions and step-wise solutions. MultiMath-7B achieves state-of-the-art (SOTA) performance among open-source models on existing multimodal mathematical benchmarks and also excels on text-only mathematical benchmarks. Our model and dataset are available at {\textcolor{blue}{\url{https://github.com/pengshuai-rin/MultiMath}}}.
This study focuses on improving the performance of lightweight Large Language Models (LLMs) in mathematical reasoning tasks. We introduce a novel method for measuring mathematical logic similarity and design an automatic screening mechanism to construct a set of reference problems that integrate both semantic and logical similarity. By employing carefully crafted positive and negative example prompts, we guide the model towards adopting sound reasoning logic. To the best of our knowledge, this is the first attempt to utilize retrieval-enhanced generation for mathematical problem-solving. Experimental results demonstrate that our method achieves a 15.8% improvement over the Chain of Thought approach on the SVAMP dataset and a 21.5 % improvement on the GSM8K dataset. Further application of this method to a large-scale model with 175 billion parameters yields performance comparable to the best results on both aforementioned datasets. Finally, we conduct an analysis of errors during the reasoning process, providing valuable insights and directions for future research on reasoning tasks using large language models.
Large language models (LLMs) have obtained promising results in mathematical reasoning, which is a foundational skill for human intelligence. Most previous studies focus on improving and measuring the performance of LLMs based on textual math reasoning datasets (e.g., MATH, GSM8K). Recently, a few researchers have released English multimodal math datasets (e.g., MATHVISTA and MATH-V) to evaluate the effectiveness of large multimodal models (LMMs). In this paper, we release a Chinese multimodal math (CMM-Math) dataset, including benchmark and training parts, to evaluate and enhance the mathematical reasoning of LMMs. CMM-Math contains over 28,000 high-quality samples, featuring a variety of problem types (e.g., multiple-choice, fill-in-the-blank, and so on) with detailed solutions across 12 grade levels from elementary to high school in China. Specifically, the visual context may be present in the questions or opinions, which makes this dataset more challenging. Through comprehensive analysis, we discover that state-of-the-art LMMs on the CMM-Math dataset face challenges, emphasizing the necessity for further improvements in LMM development. We also propose a Multimodal Mathematical LMM (Math-LMM) to handle the problems with mixed input of multiple images and text segments. We train our model using three stages, including foundational pre-training, foundational fine-tuning, and mathematical fine-tuning. The extensive experiments indicate that our model effectively improves math reasoning performance by comparing it with the SOTA LMMs over three multimodal mathematical datasets.
Recent studies have shown that large language models' (LLMs) mathematical problem-solving capabilities can be enhanced by integrating external tools, such as code interpreters, and employing multi-turn Chain-of-Thought (CoT) reasoning. While current methods focus on synthetic data generation and Supervised Fine-Tuning (SFT), this paper studies the complementary direct preference learning approach to further improve model performance. However, existing direct preference learning algorithms are originally designed for the single-turn chat task, and do not fully address the complexities of multi-turn reasoning and external tool integration required for tool-integrated mathematical reasoning tasks. To fill in this gap, we introduce a multi-turn direct preference learning framework, tailored for this context, that leverages feedback from code interpreters and optimizes trajectory-level preferences. This framework includes multi-turn DPO and multi-turn KTO as specific implementations. The effectiveness of our framework is validated through training of various language models using an augmented prompt set from the GSM8K and MATH datasets. Our results demonstrate substantial improvements: a supervised fine-tuned Gemma-1.1-it-7B model's performance increased from 77.5% to 83.9% on GSM8K and from 46.1% to 51.2% on MATH. Similarly, a Gemma-2-it-9B model improved from 84.1% to 86.3% on GSM8K and from 51.0% to 54.5% on MATH.
Traditional OCR systems (OCR-1.0) are increasingly unable to meet people's usage due to the growing demand for intelligent processing of man-made optical characters. In this paper, we collectively refer to all artificial optical signals (e.g., plain texts, math/molecular formulas, tables, charts, sheet music, and even geometric shapes) as "characters" and propose the General OCR Theory along with an excellent model, namely GOT, to promote the arrival of OCR-2.0. The GOT, with 580M parameters, is a unified, elegant, and end-to-end model, consisting of a high-compression encoder and a long-contexts decoder. As an OCR-2.0 model, GOT can handle all the above "characters" under various OCR tasks. On the input side, the model supports commonly used scene- and document-style images in slice and whole-page styles. On the output side, GOT can generate plain or formatted results (markdown/tikz/smiles/kern) via an easy prompt. Besides, the model enjoys interactive OCR features, i.e., region-level recognition guided by coordinates or colors. Furthermore, we also adapt dynamic resolution and multi-page OCR technologies to GOT for better practicality. In experiments, we provide sufficient results to prove the superiority of our model.
Self-correction is a novel method that can stimulate the potential reasoning abilities of large language models (LLMs). It involves detecting and correcting errors during the inference process when LLMs solve reasoning problems. However, recent works do not regard self-correction as a spontaneous and intrinsic capability of LLMs. Instead, such correction is achieved through post-hoc generation, external knowledge introduction, multi-model collaboration, and similar techniques. In this paper, we propose a series of mathematical LLMs called S$^3$c-Math, which are able to perform Spontaneous Step-level Self-correction for Mathematical reasoning. This capability helps LLMs to recognize whether their ongoing inference tends to contain errors and simultaneously correct these errors to produce a more reliable response. We proposed a method, which employs a step-level sampling approach to construct step-wise self-correction data for achieving such ability. Additionally, we implement a training strategy that uses above constructed data to equip LLMs with spontaneous step-level self-correction capacities. Our data and methods have been demonstrated to be effective across various foundation LLMs, consistently showing significant progress in evaluations on GSM8K, MATH, and other mathematical benchmarks. To the best of our knowledge, we are the first to introduce the spontaneous step-level self-correction ability of LLMs in mathematical reasoning.
Large Language Models (LLMs) have demonstrated remarkable efficiency in tackling various tasks based on human instructions, but studies reveal that they often struggle with tasks requiring reasoning, such as math or physics. This limitation raises questions about whether LLMs truly comprehend embedded knowledge or merely learn to replicate the token distribution without a true understanding of the content. In this paper, we delve into this problem and aim to enhance the reasoning capabilities of LLMs. First, we investigate if the model has genuine reasoning capabilities by visualizing the text generation process at the attention and representation level. Then, we formulate the reasoning process of LLMs into a causal framework, which provides a formal explanation of the problems observed in the visualization. Finally, building upon this causal framework, we propose Deconfounded Causal Adaptation (DCA), a novel parameter-efficient fine-tuning (PEFT) method to enhance the model's reasoning capabilities by encouraging the model to extract the general problem-solving skills and apply these skills to different questions. Experiments show that our method outperforms the baseline consistently across multiple benchmarks, and with only 1.2M tunable parameters, we achieve better or comparable results to other fine-tuning methods. This demonstrates the effectiveness and efficiency of our method in improving the overall accuracy and reliability of LLMs.
In this paper, we investigate the presence of additive bias in Large Language Models (LLMs), drawing a parallel to the cognitive bias observed in humans where individuals tend to favor additive over subtractive changes. Using a series of controlled experiments, we tested various LLMs, including GPT-3.5 Turbo, Claude 3.5 Sonnet, Mistral, Math$\Sigma$tral, and Llama 3.1, on tasks designed to measure their propensity for additive versus subtractive modifications. Our findings demonstrate a significant preference for additive changes across all tested models. For example, in a palindrome creation task, Llama 3.1 favored adding letters 97.85% of the time over removing them. Similarly, in a Lego tower balancing task, GPT-3.5 Turbo chose to add a brick 76.38% of the time rather than remove one. In a text summarization task, Mistral 7B produced longer summaries in 59.40% to 75.10% of cases when asked to improve its own or others' writing. These results indicate that, similar to humans, LLMs exhibit a marked additive bias, which might have implications when LLMs are used on a large scale. Addittive bias might increase resource use and environmental impact, leading to higher economic costs due to overconsumption and waste. This bias should be considered in the development and application of LLMs to ensure balanced and efficient problem-solving approaches.
In this paper, we propose Singular Values and Orthonormal Regularized Singular Vectors Adaptation, or SORSA, a novel PEFT method. Each SORSA adapter consists of two main parts: trainable principal singular weights $W_p = U_p \text{diag}(S_p) V^\top_p$, and frozen residual weights $W_r = U_r \text{diag}(S_r) V^\top_r$. These parts are initialized by performing singular value decomposition (SVD) on pre-trained weights. Moreover, we implement and analyze an orthonormal regularizer, which we prove could decrease the condition number of $W_p$ and make the optimization more efficient. SORSA adapters could be merged during inference, thus eliminating any inference latency. We also introduce a method to analyze the variation of the parameters by performing SVD and discuss and analyze SORSA's superiority in minimizing the alteration in the SVD aspect. After all, SORSA shows a faster convergence than LoRA and PiSSA in our experiments. On the GSM-8K benchmark, Llama 2 7B adapted using SORSA achieved 56.03% accuracy, surpassing LoRA (42.30%), AdaLoRA (47.30%), Full FT (49.05%), and PiSSA (53.07%). On the MATH benchmark, SORSA achieved 10.36% accuracy, outperforming LoRA (5.50%), AdaLoRA (6.48%), Full FT (7.22%), and PiSSA (7.44%). We conclude that SORSA offers a new perspective on parameter-efficient fine-tuning, demonstrating remarkable performance.
People with visual impairments often struggle to create content that relies heavily on visual elements, particularly when conveying spatial and structural information. Existing accessible drawing tools, which construct images line by line, are suitable for simple tasks like math but not for more expressive artwork. On the other hand, emerging generative AI-based text-to-image tools can produce expressive illustrations from descriptions in natural language, but they lack precise control over image composition and properties. To address this gap, our work integrates generative AI with a constructive approach that provides users with enhanced control and editing capabilities. Our system, AltCanvas, features a tile-based interface enabling users to construct visual scenes incrementally, with each tile representing an object within the scene. Users can add, edit, move, and arrange objects while receiving speech and audio feedback. Once completed, the scene can be rendered as a color illustration or as a vector for tactile graphic generation. Involving 14 blind or low-vision users in design and evaluation, we found that participants effectively used the AltCanvas workflow to create illustrations.
Self-critic has become a crucial mechanism for enhancing the reasoning performance of LLMs. However, current approaches mainly involve basic prompts for intuitive instance-level feedback, which resembles System-1 processes and limits the reasoning capabilities. Moreover, there is a lack of in-depth investigations into the relationship between LLM's ability to criticize and its task-solving performance. To address these issues, we propose Critic-CoT, a novel framework that pushes LLMs toward System-2-like critic capability. Through a step-wise CoT reasoning paradigm and the automatic construction of distant-supervision data without human annotation, Critic-CoT enables LLMs to engage in slow, analytic self-critique and refinement, thereby improving their reasoning abilities. Experiments on GSM8K and MATH demonstrate that our enhanced model significantly boosts task-solving performance by filtering out invalid solutions or iterative refinement. Furthermore, we investigate the intrinsic correlation between critique and task-solving abilities within LLMs, discovering that these abilities can mutually reinforce each other rather than conflict.
Language models have demonstrated remarkable performance in solving reasoning tasks; however, even the strongest models still occasionally make reasoning mistakes. Recently, there has been active research aimed at improving reasoning accuracy, particularly by using pretrained language models to "self-correct" their mistakes via multi-round prompting. In this paper, we follow this line of work but focus on understanding the usefulness of incorporating "error-correction" data directly into the pretraining stage. This data consists of erroneous solution steps immediately followed by their corrections. Using a synthetic math dataset, we show promising results: this type of pretrain data can help language models achieve higher reasoning accuracy directly (i.e., through simple auto-regression, without multi-round prompting) compared to pretraining on the same amount of error-free data. We also delve into many details, such as (1) how this approach differs from beam search, (2) how such data can be prepared, (3) whether masking is needed on the erroneous tokens, (4) the amount of error required, (5) whether such data can be deferred to the fine-tuning stage, and many others.
Large language models rely on Supervised Fine-Tuning (SFT) to specialize in downstream tasks. Cross Entropy (CE) loss is the de facto choice in SFT, but it often leads to overfitting and limited output diversity due to its aggressive updates to the data distribution. This paper aim to address these issues by introducing the maximum entropy principle, which favors models with flatter distributions that still effectively capture the data. Specifically, we develop a new distribution matching method called GEM, which solves reverse Kullback-Leibler divergence minimization with an entropy regularizer. For the SFT of Llama-3-8B models, GEM outperforms CE in several aspects. First, when applied to the UltraFeedback dataset to develop general instruction-following abilities, GEM exhibits reduced overfitting, evidenced by lower perplexity and better performance on the IFEval benchmark. Furthermore, GEM enhances output diversity, leading to performance gains of up to 7 points on math reasoning and code generation tasks using best-of-n sampling, even without domain-specific data. Second, when fine-tuning with domain-specific datasets for math reasoning and code generation, GEM also shows less overfitting and improvements of up to 10 points compared with CE.
There is a growing trend of teaching large language models (LLMs) to solve mathematical problems through coding. Existing studies primarily focus on prompting powerful, closed-source models to generate seed training data followed by in-domain data augmentation, equipping LLMs with considerable capabilities for code-aided mathematical reasoning. However, continually training these models on augmented data derived from a few datasets such as GSM8K may impair their generalization abilities and restrict their effectiveness to a narrow range of question types. Conversely, the potential of improving such LLMs by leveraging large-scale, expert-written, diverse math question-answer pairs remains unexplored. To utilize these resources and tackle unique challenges such as code response assessment, we propose a novel paradigm that uses a code-based critic model to guide steps including question-code data construction, quality control, and complementary evaluation. We also explore different alignment algorithms with self-generated instruction/preference data to foster continuous improvement. Experiments across both in-domain (up to +5.7%) and out-of-domain (+4.4%) benchmarks in English and Chinese demonstrate the effectiveness of the proposed paradigm.
Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.
Verifiers or reward models are often used to enhance the reasoning performance of large language models (LLMs). A common approach is the Best-of-N method, where N candidate solutions generated by the LLM are ranked by a verifier, and the best one is selected. While LLM-based verifiers are typically trained as discriminative classifiers to score solutions, they do not utilize the text generation capabilities of pretrained LLMs. To overcome this limitation, we instead propose training verifiers using the ubiquitous next-token prediction objective, jointly on verification and solution generation. Compared to standard verifiers, such generative verifiers (GenRM) can benefit from several advantages of LLMs: they integrate seamlessly with instruction tuning, enable chain-of-thought reasoning, and can utilize additional test-time compute via majority voting for better verification. We demonstrate that GenRM outperforms discriminative, DPO verifiers, and LLM-as-a-Judge, resulting in a 16-40% improvement in the number of problems solved with Best-of-N on algorithmic and math reasoning tasks. Furthermore, we find that training GenRM with synthetic verification rationales is sufficient to pick out subtle errors on math problems. Finally, we demonstrate that generative verifiers scale favorably with model size and inference-time compute.
Mathematical reasoning is an important research direction in the field of artificial intelligence. This article proposes a novel multi tool application framework for mathematical reasoning, aiming to achieve more comprehensive and accurate mathematical reasoning by utilizing the collaborative effect of large language models (LLMs) and multiple external tools. Firstly, use a Math Tool to perform basic mathematical calculations during the inference process through interaction with LLM. Secondly, Code Tool can generate code fragments that comply with syntax rules and execute them, providing support for complex mathematical problems. Then, through the iterative reasoning of the CoT Tool, the logical coherence and accuracy of mathematical reasoning are enhanced. Ultimately, by using self consistency tools to select the final answer based on different parameters, the consistency and reliability of reasoning are improved. Through the synergistic effect of these tools, the framework has achieved significant performance improvement in mathematical reasoning tasks. We conducted experiments on the NumGLUE Task 4 test set, which includes 220 mathematical reasoning fill in the blank questions. The experimental results showed that, based on Math Tool, Code Tool, and CoT Tool, in Task 4 task,our method achieved an accuracy of 89.09,compared with the GPT3+FewShot baseline, Few Shot+ERNIE-4.0+self consistency improved by 49.09%, and compared with fine-tuning the Fine tuning baseline, Few Shot+ERNIE-4.0+self consistency improved by 52.29%
Mathematical information is essential for technical work, but its creation, interpretation, and search are challenging. To help address these challenges, researchers have developed multimodal search engines and mathematical question answering systems. This book begins with a simple framework characterizing the information tasks that people and systems perform as we work to answer math-related questions. The framework is used to organize and relate the other core topics of the book, including interactions between people and systems, representing math formulas in sources, and evaluation. We close by addressing some key questions and presenting directions for future work. This book is intended for students, instructors, and researchers interested in systems that help us find and use mathematical information.
The use of Large Language Models (LLMs) in mathematical reasoning has become a cornerstone of related research, demonstrating the intelligence of these models and enabling potential practical applications through their advanced performance, such as in educational settings. Despite the variety of datasets and in-context learning algorithms designed to improve the ability of LLMs to automate mathematical problem solving, the lack of comprehensive benchmarking across different datasets makes it complicated to select an appropriate model for specific tasks. In this project, we present a benchmark that fairly compares seven state-of-the-art in-context learning algorithms for mathematical problem solving across five widely used mathematical datasets on four powerful foundation models. Furthermore, we explore the trade-off between efficiency and performance, highlighting the practical applications of LLMs for mathematical reasoning. Our results indicate that larger foundation models like GPT-4o and LLaMA 3-70B can solve mathematical reasoning independently from the concrete prompting strategy, while for smaller models the in-context learning approach significantly influences the performance. Moreover, the optimal prompt depends on the chosen foundation model. We open-source our benchmark code to support the integration of additional models in future research.
Prompt engineering has demonstrated remarkable success in enhancing the performance of large language models (LLMs) across diverse tasks. However, most existing prompt optimization methods only focus on the task-level performance, overlooking the importance of query-preferred prompts, which leads to suboptimal performances. Additionally, these methods rely heavily on frequent interactions with LLMs to obtain feedback for guiding the optimization process, incurring substantial redundant interaction costs. In this paper, we introduce Query-dependent Prompt Optimization (QPO), which leverages multi-loop offline reinforcement learning to iteratively fine-tune a small pretrained language model to generate optimal prompts tailored to the input queries, thus significantly improving the prompting effect on the large target LLM. We derive insights from offline prompting demonstration data, which already exists in large quantities as a by-product of benchmarking diverse prompts on open-sourced tasks, thereby circumventing the expenses of online interactions. Furthermore, we continuously augment the offline dataset with the generated prompts in each loop, as the prompts from the fine-tuned model are supposed to outperform the source prompts in the original dataset. These iterative loops bootstrap the model towards generating optimal prompts. Experiments on various LLM scales and diverse NLP and math tasks demonstrate the efficacy and cost-efficiency of our method in both zero-shot and few-shot scenarios.