Neuro-Symbolic Data Generation for Math Reasoning

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TL;DR

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Key Takeaways

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Why does it matter?

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This paper is crucial for researchers in AI and mathematics because it directly addresses the critical issue of LLM limitations in mathematical reasoning. By introducing a novel neuro-symbolic data generation method, the study opens up new avenues for improving LLM performance in this domain. The proposed approach’s effectiveness and scalability make it highly relevant to the current research trends in augmenting LLMs with external knowledge and improving their reasoning abilities. Furthermore, the generated dataset offers a valuable resource for future research in mathematical problem solving with LLMs.

Visual Insights

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This figure illustrates the three main steps of the neuro-symbolic data generation framework: formalization of a seed problem into a symbolic representation, mutation of the symbolic problem to create diverse variants, and informalization of the mutated symbolic problems back into natural language. The process leverages both the strengths of LLMs (intuitive informalization) and math solvers (precise symbolic reasoning). GPT-4 is used to generate reasoning paths, which are then verified by symbolic solvers to ensure validity.

This table presents the consistency rates of different operations used in the informalization process of converting formal math problems into natural language. The operations include mutation, few-shot learning, comment generation, math-word instruction, problem modification, and variable refresh. Two recommended patterns (P1 and P2) combine these operations, showing improved consistency rates compared to the base case.

In-depth insights

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Neuro-symbolic Approach

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A neuro-symbolic approach integrates the strengths of neural networks and symbolic reasoning to overcome limitations of each. Neural networks excel at learning complex patterns from data but lack explainability and struggle with symbolic manipulation. Conversely, symbolic methods provide explicit representations and logical reasoning but require extensive hand-crafted knowledge and are limited in their ability to handle noisy or incomplete data. Neuro-symbolic approaches aim to combine these strengths by using neural networks to learn features or patterns relevant to a symbolic system, then using symbolic manipulation to perform reasoning and generate predictions. This hybrid approach enables systems that are both powerful and interpretable, particularly beneficial for tasks that necessitate both pattern recognition and logical inference, like solving mathematical problems or performing commonsense reasoning. Key challenges include effective integration of the two paradigms and the development of robust training methods.

MCMC Mutation

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The concept of “MCMC Mutation” within a research paper likely refers to a method for generating diverse and valid mathematical problems. It combines the strengths of Markov Chain Monte Carlo (MCMC) sampling with symbolic manipulation. MCMC’s role is to explore the vast space of possible problem variations, generating diverse mutations of a seed problem. However, random mutations can easily invalidate a math problem, leading to unsolvable or nonsensical questions. This is where symbolic solvers become critical. Symbolic solvers check the validity of each mutated problem, ensuring that the generated problems remain mathematically sound. The combination of MCMC for exploration and symbolic solvers for validation is crucial to the success of this data augmentation strategy, enabling the creation of a large and high-quality dataset for training machine learning models in mathematical reasoning.

LLM Fine-tuning

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LLM fine-tuning in the context of mathematical reasoning presents a unique challenge. Standard fine-tuning approaches might not suffice due to the scarcity of high-quality mathematical datasets. The paper emphasizes the need for a framework that generates diverse yet valid data, thus highlighting the crucial role of neuro-symbolic techniques. By combining the strengths of LLMs for intuitive problem informalization with the precision of symbolic solvers for validation, a more effective fine-tuning process is achieved. The results demonstrate improved performance across multiple benchmarks, outperforming existing open-source methods. However, limitations exist concerning the capabilities of symbolic solvers and the dependence on GPT-4, suggesting areas for future improvement and exploration of alternative methods for data generation and informalization.

Data Efficiency

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The concept of data efficiency is central to evaluating the success of the neuro-symbolic framework introduced in this research. The study demonstrates that the framework generates high-quality mathematical datasets, surpassing existing methods in terms of both diversity and validity. This superior data quality leads to significant improvements in the performance of LLMs fine-tuned using the generated datasets. However, a deeper analysis is needed to fully quantify data efficiency. The study mentions comparing the generated datasets’ size to existing benchmarks, yet it doesn’t provide a clear metric for comparing performance gains against the amount of data used. This leaves room for future research to establish a more precise benchmark for evaluating data efficiency in the context of neuro-symbolic mathematical reasoning. Furthermore, exploring the trade-off between data quality and quantity to determine the most efficient balance is essential. Investigating the scalability of the framework with larger datasets is key, as a more efficient data generation method should demonstrate consistent performance gains even with increased data scales.

Future Research

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Future research directions stemming from this neuro-symbolic data generation framework for mathematical reasoning could involve expanding the expressiveness of the mutation operators to create even more diverse and challenging problems, potentially incorporating techniques like problem fusion to combine simpler problems into more complex ones. Another critical area is improving the capabilities of symbolic solvers by enhancing their ability to handle higher-order mathematical concepts and inequalities, perhaps through integration with advanced mathematical software or novel algorithmic approaches. Additionally, reducing the reliance on GPT-4 for informalization and reasoning path generation could be explored through curriculum learning or fine-tuning specialized LLMs, improving efficiency and robustness. Finally, a significant focus should be placed on exploring the generalizability of the method to other domains beyond mathematics, assessing its applicability and effectiveness across a broader range of symbolic reasoning tasks.

More visual insights

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More on figures

This figure shows the results of experiments evaluating the proposed mutation mechanism for generating math problems with varying difficulty levels. The left sub-figure displays the distribution of reasoning steps required by GPT-4 to solve problems at different difficulty levels, indicating that more difficult problems necessitate more reasoning steps. The right sub-figure illustrates the relationship between the number of reasoning steps in training and the reasoning error rate during testing. The results demonstrate that incorporating more complex problems into the training data steadily improves the LLM’s reasoning capabilities.

This figure presents the BLEU scores comparing the model’s generated solutions against ground truth and GPT-4 solutions on both the training and test sets of the GSM8K and MATH datasets. The low BLEU scores on the test sets for both our model and MetaMathQA demonstrate that neither model is memorizing training data, thus proving the data generation method doesn’t introduce data contamination.

This figure compares the performance of models fine-tuned using datasets generated by the proposed method and MetaMathQA. The x-axis represents the size of the training dataset, and the y-axis shows the accuracy on four different benchmark datasets (GSM8K, SVAMP, ASDiv, and MATH). The lines show that increased training data consistently improves model performance across all datasets, and that the proposed method generally outperforms MetaMathQA.

This figure illustrates the three main steps of the neuro-symbolic data generation framework: formalization of a seed problem into its symbolic representation, mutation of the symbolic problem to generate diverse and valid variants, and informalization of the mutated symbolic problems back into natural language using LLMs. GPT-4 is used to generate reasoning paths which are verified by symbolic solvers to ensure validity and consistency.

This figure shows the diversity gain (a measure of how different the generated data is from the original dataset) at different difficulty levels and data budget sizes. The diversity is calculated using the BERT model as a feature extractor. The results show that increasing the size of the dataset consistently enhances the diversity, with a mix of difficulty levels yielding the highest gains.

More on tables

This table compares the performance of the proposed neuro-symbolic data generation method against other state-of-the-art (SOTA) large language models (LLMs) on two benchmark datasets: GSM8K and MATH. Three different base LLMs (LLaMA-2 7B, LLaMA-2 13B, and Mistral 7B) were fine-tuned using the generated datasets. The table shows the accuracy of each model on each dataset, highlighting the best performance in bold, and providing the difference in performance between the proposed model and other SOTA LLMs. This demonstrates the effectiveness of the proposed method in improving the mathematical reasoning capabilities of LLMs.

This table compares the performance of the proposed neuro-symbolic data generation method against other state-of-the-art (SOTA) large language models (LLMs) on two mathematical reasoning datasets (GSM8K and MATH). Three different base LLMs (LLaMA-2 7B, LLaMA-13B, and Mistral 7B) were fine-tuned using the generated data. The table shows the accuracy of each model on both datasets and highlights the improvement achieved by the proposed method over the SOTA models.

This table compares the performance of the proposed neuro-symbolic data generation method against other state-of-the-art (SOTA) Large Language Models (LLMs) on two mathematical reasoning datasets: GSM8K and MATH. Three different base LLMs (LLaMA-2 7B, LLaMA-13B, and Mistral 7B) were fine-tuned using the generated datasets, and their performance is compared against SOTA models like WizardMath, MuggleMATH, MAmmoTH, and MetaMath. The table highlights the accuracy achieved by each model on both datasets and shows the improvement achieved by the proposed method over the other SOTA models.

This table compares the performance of the proposed method and MetaMath on the MATH dataset across seven different mathematical categories. It shows the accuracy achieved by each method in each category and the improvement achieved by the proposed method over MetaMath. The base model used for both methods is Mistral-7B. The best performing model for each category is highlighted in bold.

This table compares the performance of the proposed model against other state-of-the-art (SOTA) large language models (LLMs) on two mathematical reasoning benchmark datasets (GSM8K and MATH). It shows the accuracy achieved by each model after fine-tuning on different base models (LLaMA-2 7B, LLaMA-2 13B, and Mistral 7B), highlighting the superior performance of the proposed method. The delta (difference) between the proposed model’s accuracy and those of the other LLMs is also given to show the extent of the improvement.

This table compares the performance of different mathematical reasoning models, including the models fine-tuned using the proposed neuro-symbolic data generation framework, across three different base models (LLaMA-2 7B, LLaMA-13B, and Mistral 7B) and two benchmark datasets (GSM8K and MATH). The best performance for each model and dataset is highlighted in bold, and the improvement achieved by the proposed method compared to other state-of-the-art (SOTA) models is also shown.

This table compares the performance of the proposed neuro-symbolic data generation method against other state-of-the-art (SOTA) large language models (LLMs) on two mathematical reasoning datasets (GSM8K and MATH). Three different base LLMs (LLaMA-2 7B, LLaMA-2 13B, and Mistral 7B) were fine-tuned using the generated data. The table shows the accuracy of each model on each dataset, highlighting the best performance in bold. It also provides the difference in performance between the proposed method and the SOTA LLMs.

Full paper

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