Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs

Siyuan Guo, Aniket Didolkar, Nan Rosemary Ke, Anirudh Goyal, Ferenc Huszár, Bernhard Schölkopf·May 24, 2024

Summary

This paper investigates the mathematical understanding of large language models (LLMs) in their capacity as scientific assistants. It distinguishes between pre-trained knowledge and in-context or instruction-tuning abilities, using the NTKEval method, inspired by the Neural Tangent Kernel. The study finds that in-context learning exhibits signs of domain understanding, particularly in differentiating deep math skills from surface structures, while instruction-tuning may not demonstrate the same level of skill-specific comprehension. The KhanSkill dataset is introduced to analyze learning alignment with human skills, and the results suggest that LLMs can adapt and learn through observation, but instruction-tuning may rely more on format matching. The research highlights the need for evaluating LLMs' ability to understand and apply math concepts for problem-solving, beyond simple pattern matching, and emphasizes the importance of considering learning-to-learn aspects in their assessment.

Paper digest

What problem does the paper attempt to solve? Is this a new problem?

The paper aims to assess mathematical understanding in Large Language Models (LLMs) beyond pattern matching by measuring mathematical problem-solving with the math dataset . This paper addresses the need to evaluate LLMs' abilities in mathematical problem-solving, focusing on understanding mathematical concepts rather than solely relying on pattern matching . The research explores the extent to which LLMs can demonstrate proficiency in mathematical reasoning and problem-solving tasks, indicating a novel approach to evaluating LLMs' mathematical understanding .


What scientific hypothesis does this paper seek to validate?

This paper aims to validate the scientific hypothesis related to advancing the field of Machine Learning by focusing on mathematical understanding in Large Language Models (LLMs) . The goal is to go beyond pattern matching and contribute to the design of improved and more transparent scientific assistants .


What new ideas, methods, or models does the paper propose? What are the characteristics and advantages compared to previous methods?

The paper "Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs" proposes several new ideas, methods, and models in the field of Machine Learning and language models . Some of the key contributions include:

  1. Emergent Abilities of Large Language Models: The paper explores the emergent abilities of large language models (LLMs) . It discusses how these models can exhibit reasoning skills through chain-of-thought prompting, which elicits reasoning in LLMs .

  2. Skills-in-Context Prompting: The paper introduces the concept of skills-in-context prompting, which aims to unlock compositionality in large language models . This method helps in understanding and training language models by providing prompts that require specific skills to be applied.

  3. Generalized Neural Tangent Kernel Analysis: The paper presents a generalized neural tangent kernel analysis for two-layer neural networks . This analysis contributes to understanding the inductive bias of neural tangent kernels, which is crucial for neural network convergence and generalization.

  4. Fine-Tuned Language Models as Zero-Shot Learners: The paper discusses how fine-tuned language models can act as zero-shot learners . This concept highlights the ability of language models to learn and perform tasks without explicit training data, showcasing their adaptability and versatility.

  5. Mathematical Discoveries from Program Search: The paper explores mathematical discoveries from program search using large language models . This approach leverages the capabilities of LLMs to generate functional protein sequences across diverse families, showcasing their potential in scientific research and discovery.

Overall, the paper introduces innovative approaches such as skills-in-context prompting, generalized neural tangent kernel analysis, and zero-shot learning capabilities of fine-tuned language models, contributing to the advancement of Machine Learning and language model research . The paper "Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs" introduces several characteristics and advantages compared to previous methods in the field of Machine Learning and language models :

  1. Skills-in-Context Prompting: The paper proposes the concept of skills-in-context prompting, which focuses on unlocking compositionality in large language models (LLMs) . This method aims to enhance the understanding and training of language models by providing prompts that require specific skills to be applied, thereby improving the model's ability to perform tasks that involve reasoning and complex problem-solving.

  2. Generalized Neural Tangent Kernel Analysis: The paper presents a generalized neural tangent kernel analysis for two-layer neural networks . This analysis contributes to understanding the inductive bias of neural tangent kernels, which is essential for neural network convergence and generalization. By exploring the neural tangent kernel, the paper enhances the understanding of the underlying mechanisms of neural networks.

  3. Zero-Shot Learning Capabilities: The paper discusses how fine-tuned language models can act as zero-shot learners . This characteristic highlights the adaptability and versatility of language models, enabling them to learn and perform tasks without explicit training data. By leveraging zero-shot learning capabilities, language models can demonstrate improved performance across various tasks and domains.

  4. Mathematical Discoveries from Program Search: The paper explores mathematical discoveries from program search using large language models . By harnessing the capabilities of LLMs, the paper showcases the potential of these models in generating functional protein sequences across diverse families, leading to advancements in scientific research and discovery.

  5. Efficiency and Accuracy: The paper emphasizes the importance of sample efficiency in evaluating models . By utilizing NTKEval, the paper measures the difference in probability of generating correct solutions between a model trained on skill-focused datasets and the base model. This approach ensures accurate assessments of model performance and enhances the efficiency of model evaluation processes.

Overall, the characteristics and advantages presented in the paper contribute to advancing the capabilities of large language models in understanding mathematical concepts, reasoning, and problem-solving tasks . The innovative methods introduced in the paper pave the way for improved model performance, efficiency, and adaptability in various applications within the field of Machine Learning and language modeling.


Do any related researches exist? Who are the noteworthy researchers on this topic in this field?What is the key to the solution mentioned in the paper?

Several related research papers exist in the field of mathematical understanding in Large Language Models (LLMs). Noteworthy researchers in this area include Siyuan Guo, Aniket Didolkar, Nan Rosemary Ke, Anirudh Goyal, Ferenc Huszár, and Bernhard Schölkopf . These researchers have contributed to assessing the domain knowledge of LLMs and understanding how these models learn to solve mathematical problems.

The key to the solution mentioned in the paper "Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs" involves proposing NTKEval, which assesses changes in LLMs' probability distribution via training on different types of mathematical data. This systematic analysis aims to evaluate the domain understanding of LLMs during in-context learning, highlighting the importance of exploiting the complex knowledge structure within mathematics .


How were the experiments in the paper designed?

The experiments in the paper were designed by evaluating the models on Code Llama 7b, Llemma 7b, and either Mistral 7b or Mixtral 8x7b Instruct . These experiments utilized a suite of Large Language Models (LLMs) tailored for code, mathematics, and general-purpose chat models to test the domain understanding of specialized models . The choice of open-sourced models allowed for both inference and instruction-tuning on a single GPU . The dataset used for evaluation consisted of 1240 questions in the training set and 620 questions evenly split across skills in the test set . The accuracy of the models was recorded and evaluated based on the experiments conducted .


What is the dataset used for quantitative evaluation? Is the code open source?

The dataset used for quantitative evaluation is the KhanSkill dataset, which consists of questions generated from the Khan Academy exercises . The Khan Exercises framework is MIT licensed, and the exercises are under a Creative Commons by-nc-sa license . This indicates that the code for the exercises is open source.


Do the experiments and results in the paper provide good support for the scientific hypotheses that need to be verified? Please analyze.

The experiments and results presented in the paper provide substantial support for the scientific hypotheses that need to be verified. The study evaluates mathematical understanding in Large Language Models (LLMs) through experiments involving different mathematical operations and skills . The research assesses accuracy differences and probabilities with varying generations per test question, providing detailed insights into the performance of LLMs in mathematical problem-solving tasks . Additionally, the paper references prior works in the field of Machine Learning and Neural Networks, indicating a comprehensive review of related research to support the scientific hypotheses . The inclusion of specific models like Code Llama, Llemma, Mistral, and Mixtral in the experiments demonstrates a focused evaluation of LLMs across different domains, enhancing the robustness of the study . Overall, the meticulous analysis of accuracy, probabilities, and model choices in the experiments contributes significantly to verifying the scientific hypotheses related to mathematical understanding in LLMs .


What are the contributions of this paper?

The contributions of the paper "Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs" include advancing the field of Machine Learning with the goal of designing better and more transparent scientific assistants . The work presented in the paper aims to enhance mathematical understanding in Large Language Models (LLMs) . The paper does not specifically highlight any societal consequences of the work .


What work can be continued in depth?

Further research in the field of Machine Learning can be extended to explore the emergent abilities of large language models (LLMs) . This includes investigating how LLMs can develop reasoning skills through chain-of-thought prompting . Additionally, there is potential for studying the capabilities of LLMs as optimizers . Moreover, the understanding of the inductive bias of neural tangent kernels can be further explored to enhance the generalization of neural networks . Further investigations can also focus on the learning-to-learn ability of LLMs when exposed to different math skills, which can contribute to improving their domain understanding and performance .


Introduction
Background
Emergence of large language models in scientific domains
Importance of understanding their cognitive capabilities
Objective
To analyze LLMs' mathematical understanding and problem-solving abilities
Differentiate pre-trained knowledge from in-context and instruction-tuning
Methodology
Data Collection
NTKEval Method
Use of Neural Tangent Kernel-inspired NTKEval for assessment
Comparison of pre-trained and fine-tuned models
Data Preprocessing
KhanSkill Dataset
Collection and preparation of math problems and human skill annotations
Experiment Design
In-context learning vs. instruction-tuning analysis
In-Context Learning Analysis
Signs of Domain Understanding
Differentiation of deep math skills from surface structures
Examples of problem-solving instances
Learning Alignment with Human Skills
KhanSkill dataset application
Observational learning and adaptation
Instruction-Tuning Evaluation
Skill-Specific Comprehension
Lack of depth in understanding compared to in-context learning
Format matching reliance
Limitations and Insights
Overreliance on pattern matching in instruction-tuning
Assessing Problem-Solving Abilities
Importance of concept understanding and application
Real-world problem-solving scenarios
Learning-to-Learn Aspects
The role of adaptation and observation in LLM performance
Future directions for evaluating LLMs' cognitive growth
Conclusion
Summary of findings and implications for LLM development
Recommendations for improved evaluation and benchmarking in math assistance
Basic info
papers
computation and language
machine learning
artificial intelligence
Advanced features
Insights
What does the study suggest about the problem-solving capabilities of LLMs through instruction-tuning?
What method does the paper use to evaluate the mathematical understanding of LLMs?
What is the significance of the KhanSkill dataset in the research?
How does the study differentiate between pre-trained knowledge and in-context learning abilities of LLMs?

Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs

Siyuan Guo, Aniket Didolkar, Nan Rosemary Ke, Anirudh Goyal, Ferenc Huszár, Bernhard Schölkopf·May 24, 2024

Summary

This paper investigates the mathematical understanding of large language models (LLMs) in their capacity as scientific assistants. It distinguishes between pre-trained knowledge and in-context or instruction-tuning abilities, using the NTKEval method, inspired by the Neural Tangent Kernel. The study finds that in-context learning exhibits signs of domain understanding, particularly in differentiating deep math skills from surface structures, while instruction-tuning may not demonstrate the same level of skill-specific comprehension. The KhanSkill dataset is introduced to analyze learning alignment with human skills, and the results suggest that LLMs can adapt and learn through observation, but instruction-tuning may rely more on format matching. The research highlights the need for evaluating LLMs' ability to understand and apply math concepts for problem-solving, beyond simple pattern matching, and emphasizes the importance of considering learning-to-learn aspects in their assessment.
Mind map
In-context learning vs. instruction-tuning analysis
Comparison of pre-trained and fine-tuned models
Use of Neural Tangent Kernel-inspired NTKEval for assessment
Overreliance on pattern matching in instruction-tuning
Format matching reliance
Lack of depth in understanding compared to in-context learning
Observational learning and adaptation
KhanSkill dataset application
Examples of problem-solving instances
Differentiation of deep math skills from surface structures
Experiment Design
NTKEval Method
Differentiate pre-trained knowledge from in-context and instruction-tuning
To analyze LLMs' mathematical understanding and problem-solving abilities
Importance of understanding their cognitive capabilities
Emergence of large language models in scientific domains
Recommendations for improved evaluation and benchmarking in math assistance
Summary of findings and implications for LLM development
Future directions for evaluating LLMs' cognitive growth
The role of adaptation and observation in LLM performance
Real-world problem-solving scenarios
Importance of concept understanding and application
Limitations and Insights
Skill-Specific Comprehension
Learning Alignment with Human Skills
Signs of Domain Understanding
Data Preprocessing
Data Collection
Objective
Background
Conclusion
Learning-to-Learn Aspects
Assessing Problem-Solving Abilities
Instruction-Tuning Evaluation
In-Context Learning Analysis
Methodology
Introduction
Outline
Introduction
Background
Emergence of large language models in scientific domains
Importance of understanding their cognitive capabilities
Objective
To analyze LLMs' mathematical understanding and problem-solving abilities
Differentiate pre-trained knowledge from in-context and instruction-tuning
Methodology
Data Collection
NTKEval Method
Use of Neural Tangent Kernel-inspired NTKEval for assessment
Comparison of pre-trained and fine-tuned models
Data Preprocessing
KhanSkill Dataset
Collection and preparation of math problems and human skill annotations
Experiment Design
In-context learning vs. instruction-tuning analysis
In-Context Learning Analysis
Signs of Domain Understanding
Differentiation of deep math skills from surface structures
Examples of problem-solving instances
Learning Alignment with Human Skills
KhanSkill dataset application
Observational learning and adaptation
Instruction-Tuning Evaluation
Skill-Specific Comprehension
Lack of depth in understanding compared to in-context learning
Format matching reliance
Limitations and Insights
Overreliance on pattern matching in instruction-tuning
Assessing Problem-Solving Abilities
Importance of concept understanding and application
Real-world problem-solving scenarios
Learning-to-Learn Aspects
The role of adaptation and observation in LLM performance
Future directions for evaluating LLMs' cognitive growth
Conclusion
Summary of findings and implications for LLM development
Recommendations for improved evaluation and benchmarking in math assistance

Paper digest

What problem does the paper attempt to solve? Is this a new problem?

The paper aims to assess mathematical understanding in Large Language Models (LLMs) beyond pattern matching by measuring mathematical problem-solving with the math dataset . This paper addresses the need to evaluate LLMs' abilities in mathematical problem-solving, focusing on understanding mathematical concepts rather than solely relying on pattern matching . The research explores the extent to which LLMs can demonstrate proficiency in mathematical reasoning and problem-solving tasks, indicating a novel approach to evaluating LLMs' mathematical understanding .


What scientific hypothesis does this paper seek to validate?

This paper aims to validate the scientific hypothesis related to advancing the field of Machine Learning by focusing on mathematical understanding in Large Language Models (LLMs) . The goal is to go beyond pattern matching and contribute to the design of improved and more transparent scientific assistants .


What new ideas, methods, or models does the paper propose? What are the characteristics and advantages compared to previous methods?

The paper "Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs" proposes several new ideas, methods, and models in the field of Machine Learning and language models . Some of the key contributions include:

  1. Emergent Abilities of Large Language Models: The paper explores the emergent abilities of large language models (LLMs) . It discusses how these models can exhibit reasoning skills through chain-of-thought prompting, which elicits reasoning in LLMs .

  2. Skills-in-Context Prompting: The paper introduces the concept of skills-in-context prompting, which aims to unlock compositionality in large language models . This method helps in understanding and training language models by providing prompts that require specific skills to be applied.

  3. Generalized Neural Tangent Kernel Analysis: The paper presents a generalized neural tangent kernel analysis for two-layer neural networks . This analysis contributes to understanding the inductive bias of neural tangent kernels, which is crucial for neural network convergence and generalization.

  4. Fine-Tuned Language Models as Zero-Shot Learners: The paper discusses how fine-tuned language models can act as zero-shot learners . This concept highlights the ability of language models to learn and perform tasks without explicit training data, showcasing their adaptability and versatility.

  5. Mathematical Discoveries from Program Search: The paper explores mathematical discoveries from program search using large language models . This approach leverages the capabilities of LLMs to generate functional protein sequences across diverse families, showcasing their potential in scientific research and discovery.

Overall, the paper introduces innovative approaches such as skills-in-context prompting, generalized neural tangent kernel analysis, and zero-shot learning capabilities of fine-tuned language models, contributing to the advancement of Machine Learning and language model research . The paper "Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs" introduces several characteristics and advantages compared to previous methods in the field of Machine Learning and language models :

  1. Skills-in-Context Prompting: The paper proposes the concept of skills-in-context prompting, which focuses on unlocking compositionality in large language models (LLMs) . This method aims to enhance the understanding and training of language models by providing prompts that require specific skills to be applied, thereby improving the model's ability to perform tasks that involve reasoning and complex problem-solving.

  2. Generalized Neural Tangent Kernel Analysis: The paper presents a generalized neural tangent kernel analysis for two-layer neural networks . This analysis contributes to understanding the inductive bias of neural tangent kernels, which is essential for neural network convergence and generalization. By exploring the neural tangent kernel, the paper enhances the understanding of the underlying mechanisms of neural networks.

  3. Zero-Shot Learning Capabilities: The paper discusses how fine-tuned language models can act as zero-shot learners . This characteristic highlights the adaptability and versatility of language models, enabling them to learn and perform tasks without explicit training data. By leveraging zero-shot learning capabilities, language models can demonstrate improved performance across various tasks and domains.

  4. Mathematical Discoveries from Program Search: The paper explores mathematical discoveries from program search using large language models . By harnessing the capabilities of LLMs, the paper showcases the potential of these models in generating functional protein sequences across diverse families, leading to advancements in scientific research and discovery.

  5. Efficiency and Accuracy: The paper emphasizes the importance of sample efficiency in evaluating models . By utilizing NTKEval, the paper measures the difference in probability of generating correct solutions between a model trained on skill-focused datasets and the base model. This approach ensures accurate assessments of model performance and enhances the efficiency of model evaluation processes.

Overall, the characteristics and advantages presented in the paper contribute to advancing the capabilities of large language models in understanding mathematical concepts, reasoning, and problem-solving tasks . The innovative methods introduced in the paper pave the way for improved model performance, efficiency, and adaptability in various applications within the field of Machine Learning and language modeling.


Do any related researches exist? Who are the noteworthy researchers on this topic in this field?What is the key to the solution mentioned in the paper?

Several related research papers exist in the field of mathematical understanding in Large Language Models (LLMs). Noteworthy researchers in this area include Siyuan Guo, Aniket Didolkar, Nan Rosemary Ke, Anirudh Goyal, Ferenc Huszár, and Bernhard Schölkopf . These researchers have contributed to assessing the domain knowledge of LLMs and understanding how these models learn to solve mathematical problems.

The key to the solution mentioned in the paper "Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs" involves proposing NTKEval, which assesses changes in LLMs' probability distribution via training on different types of mathematical data. This systematic analysis aims to evaluate the domain understanding of LLMs during in-context learning, highlighting the importance of exploiting the complex knowledge structure within mathematics .


How were the experiments in the paper designed?

The experiments in the paper were designed by evaluating the models on Code Llama 7b, Llemma 7b, and either Mistral 7b or Mixtral 8x7b Instruct . These experiments utilized a suite of Large Language Models (LLMs) tailored for code, mathematics, and general-purpose chat models to test the domain understanding of specialized models . The choice of open-sourced models allowed for both inference and instruction-tuning on a single GPU . The dataset used for evaluation consisted of 1240 questions in the training set and 620 questions evenly split across skills in the test set . The accuracy of the models was recorded and evaluated based on the experiments conducted .


What is the dataset used for quantitative evaluation? Is the code open source?

The dataset used for quantitative evaluation is the KhanSkill dataset, which consists of questions generated from the Khan Academy exercises . The Khan Exercises framework is MIT licensed, and the exercises are under a Creative Commons by-nc-sa license . This indicates that the code for the exercises is open source.


Do the experiments and results in the paper provide good support for the scientific hypotheses that need to be verified? Please analyze.

The experiments and results presented in the paper provide substantial support for the scientific hypotheses that need to be verified. The study evaluates mathematical understanding in Large Language Models (LLMs) through experiments involving different mathematical operations and skills . The research assesses accuracy differences and probabilities with varying generations per test question, providing detailed insights into the performance of LLMs in mathematical problem-solving tasks . Additionally, the paper references prior works in the field of Machine Learning and Neural Networks, indicating a comprehensive review of related research to support the scientific hypotheses . The inclusion of specific models like Code Llama, Llemma, Mistral, and Mixtral in the experiments demonstrates a focused evaluation of LLMs across different domains, enhancing the robustness of the study . Overall, the meticulous analysis of accuracy, probabilities, and model choices in the experiments contributes significantly to verifying the scientific hypotheses related to mathematical understanding in LLMs .


What are the contributions of this paper?

The contributions of the paper "Learning Beyond Pattern Matching? Assaying Mathematical Understanding in LLMs" include advancing the field of Machine Learning with the goal of designing better and more transparent scientific assistants . The work presented in the paper aims to enhance mathematical understanding in Large Language Models (LLMs) . The paper does not specifically highlight any societal consequences of the work .


What work can be continued in depth?

Further research in the field of Machine Learning can be extended to explore the emergent abilities of large language models (LLMs) . This includes investigating how LLMs can develop reasoning skills through chain-of-thought prompting . Additionally, there is potential for studying the capabilities of LLMs as optimizers . Moreover, the understanding of the inductive bias of neural tangent kernels can be further explored to enhance the generalization of neural networks . Further investigations can also focus on the learning-to-learn ability of LLMs when exposed to different math skills, which can contribute to improving their domain understanding and performance .

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