Metacognitive Capabilities of LLMs: An Exploration in Mathematical Problem Solving

Summary

The paper investigates the presence of metacognitive knowledge in large language models (LLMs), specifically focusing on their ability to reason and apply skills in mathematical contexts. The authors develop a prompt-guided interaction procedure that enables a powerful LLM to assign skill labels to math questions and perform semantic clustering to create coarser, interpretable families of skill labels. These labels are validated through experiments where GPT-4 assigns skill labels to math questions in the GSM8K and MATH datasets. By presenting the LLM with exemplar solved questions associated with the identified skill labels, the accuracy of solving test questions improves for several strong LLMs, including code-assisted models. The methodology, though applied to math problems in this study, is designed to be domain-agnostic (but more evidence is needed to demonstrate that this is the case). The major contributions of this paper are as follows. It introduces a framework for extracting metacognitive knowledge from LLMs by categorizing questions based on the skills needed to solve them, resulting in the creation of a Skill Exemplar Repository. This repository, containing annotated question-answer pairs with respective skill labels, enhances the in-context learning capabilities of LLMs, leading to performance improvements across various mathematical datasets, from grade-level to competition-level problems. The approach shows promise even for weaker LLMs. While the current methodology assigns only one skill per math question, the authors acknowledge the need for a more advanced system to account for multiple skills. They also suggest future work to extend these methods for fine-tuning models, potentially bootstrapping their capabilities in broader contexts beyond math.

Strengths

The approach of using skill exemplars provides explicit labeling of skills, offering clear guidance that enhances targeted learning for specific tasks. This structured method mimics human problem-solving techniques by presenting congruent examples, potentially improving the LLM’s performance on tasks requiring particular skills. This can be especially beneficial in applications where skills can be clearly defined and labeled, such as educational tools or domain-specific problem-solving. Originality: The paper introduces a novel framework for extracting metacognitive knowledge from LLMs by categorizing math questions based on the skills required to solve them. This approach leverages the capabilities of LLMs, such as GPT-4, to assign skill labels and perform semantic clustering, creating interpretable families of skill labels. This originality is evident in the development of a Skill Exemplar Repository, which enhances the in-context learning capabilities of LLMs. The application of this framework to mathematical problems, while promising to be domain-agnostic, showcases a creative combination of existing ideas and methods, pushing the boundaries of what LLMs can achieve in terms of reasoning and applying skills. Quality: The quality of the research is underscored by the thoroughness of the experimental validation. The authors have conducted extensive experiments using well-known datasets such as GSM8K and MATH, demonstrating that presenting LLMs with exemplar solved questions associated with skill labels improves their accuracy in solving test questions. The use of multiple strong LLMs, including code-assisted models, adds robustness to the findings. The meticulous design of the prompt-guided interaction procedure and the validation of the skill labels through experiments further attest to the high quality of the research. Clarity: The paper is well-structured and clearly articulates the problem, methodology, and results. The step-by-step explanation of the prompt-guided interaction procedure and the creation of the Skill Exemplar Repository is detailed and easy to follow. The authors have done a commendable job in explaining how their approach can potentially be extended beyond mathematical contexts, adding clarity to the potential impact of their work. However, there are minor issues with consistency and completeness in the related work section and some references that need addressing to enhance overall clarity. Significance: The significance of the paper lies in its potential to significantly advance the capabilities of LLMs in reasoning and applying skills across various domains. By developing a framework that enhances in-context learning, the authors have shown that even weaker LLMs can benefit from this approach. This has far-reaching implications for the broader application of LLMs in educational tools, automated tutoring systems, and other areas requiring complex problem-solving capabilities. The proposed methodology opens new avenues for future research in fine-tuning models and bootstrapping their capabilities in diverse contexts.

Weaknesses

However, this approach may not scale well for complex tasks requiring multiple overlapping or non-overlapping skills or where skills are not easily categorized. It demands a highly curated and labeled dataset, which can be resource-intensive to create and maintain. Additionally, by explicitly labeling skills, the model's flexibility might be reduced, potentially limiting its ability to generalize to unseen tasks or novel problem types that do not fit neatly into predefined skill categories. The complexity of managing a repository of skill exemplars adds further implementation challenges. Specificity and Scalability: The approach might not scale well for more complex tasks that require multiple overlapping skills or where skills are not easily categorized. It also demands a highly curated and labeled dataset, which could be resource-intensive to create and maintain. This is acknowledged by the authors as a limitation of the current work. Comparison to in-context multi-shot learning. By explicitly labeling skills, the model's flexibility might be reduced, potentially limiting its ability to generalize to unseen tasks or novel problem types that do not fit neatly into predefined skill categories. The related work section is very short and the references are bulk listed. Some references are incomplete (e.g., [16], [20], [32], [33]). Please fix. Most of these are missing information about where the paper was published. If ArXiv note it, if already published, note that instead. Stray bracket ) on line 119. Stray space after GPT-4-0613 on line 135. Some acronyms are repeatedly introduced: e.g., Chain-of-Thought (CoT) [four times]; program-aided language models (PALs) [multiple times]. [32] talks of complex skills — which are composed of lower-order skills? How does the current work handle such complex hierarchical skill definitions. SKILL-MIX may be referred as well — https://arxiv.org/pdf/2310.17567. Describe how your work corresponds to a mixture of skills rather than just a single skill categorization (or at least discuss the implications of this in your work) Figure 1 uses two alternative notations: e.g., Skill 2 and Skill II. Is there a difference? If not, just use one notation consistently.

Questions

How have you determined how many coarse-grained categories to have for the skills — and how do more or less skill categories affect performance? How does the proposed framework handle tasks that require multiple overlapping skills or skills that are not easily categorized? Can you provide insights or preliminary results on how the system performs in more complex scenarios where skill categorization is challenging? Why are results not shown separately for CoT in Table 1? Could CoT not be applied separately to these kinds of questions. How would the results be (let’s say in Table 7), if we identify multiple skills as being relevant to the question at hand [instead of just one].

Rating

5

Confidence

4

Soundness

2

Presentation

3

Contribution

2

Limitations

In my opinion, it is a bit of a stretch to call the ability to categorize as a metacognitive ability of the LLM. I think metacognition would require many other skills as well. Can you comment on this and describe your scoping of metacognitive ability and how your work achieves significant metacognition. Metacognition traditionally refers to an individual's awareness and understanding of their own thought processes, including the ability to monitor, control, and plan their cognitive activities. However, when thinking deeply about someone else's thinking, the term typically used is theory of mind rather than metacognition. In contrast to chain of thought which better relates to metacognition, perhaps the way categorization and skill exemplars is used is closer to the theory of mind aspect. Please consider and see if you want to incorporate this into your work. “This analysis underlines the approach’s strengths in primary skill application but also its limitations in more nuanced or compound skill scenarios. It would be worthwhile to work with more complex skills.”-> I think not considering these complex skills is one of the weakness and limitation of the work. Can you do at least some preliminary analysis of complex skills to improve the completeness and contributions of this work. A significant limitation of the current methodology is its assignment of only one skill to each math question, which restricts the contributions of this work. Mathematical problems often require multiple interacting skills, and by not accounting for this complexity, the methodology can oversimplify the problem-solving process, reducing the model's generalizability and accuracy. As a suggestion for the rebuttal stage, the authors could propose implementing a more advanced method using an LLM to create skill hierarchies and assign multiple skills to each datapoint. Providing preliminary analysis or examples of this approach could strengthen the rebuttal by demonstrating its potential to enhance the model's robustness and applicability in various mathematical contexts. This proactive step would address the limitation and significantly strengthen the contributions of their work.

Summary

This paper studied how metacognitive knowledge can improve LLM's performance in two math datasets. Author asked LLM to solve math questions and identify a skills needed within a provided list of skills. The author used pedagogy research and use experiment to show improving LLM's metacognitive knowledge can improve LLM's capabilities. Asking LLM "Is this a good training datapoint for an LLM" reveals LLM's metacognitivity. The LLM will solve the problem by checking previous identified exemplars first as in-context learning examples. This paper then described the skill discovery process in 3 steps: Assign skill name -> Semantic Skill Clustering -> Skill Exemplar Repository. This paper also highlighted strong LLMs can assign intuitive skill names to questions can help weaker LLMs perform better. The proposed method shows better performance than CoT and can combine with CoT to further improve the performance. This paper highlighted one limitation of this method as only one skill are assigned to each math question. However, for more complex questions multiple skills might be involved, thus building a hierarchies of skills may further improve the results.

Strengths

1. This method viewed RAG from new angle - metacognitive capabilities of LLM. This is novel and very eye-catching as well. 2. This method does show consistent improvement over other prompting method and is versatile enough to combine with CoT. 3. Skill discovery process are fully automated using LLM. 4. Overall well written with clear logic and easy to follow.

Weaknesses

1. The method overall shows little difference over RAG idea. It's basically finding similar question LLMs have done in the past and use that for in-context learning to improve model performance. Instead of highlighting metacognitive capabilities, this is more like taking an "open-book exam" improves LLMs performance. 2. The paper highlighted Metacognitive Capabilities of LLM, but has not been discussed extensively. It's not clear weather LLMs have metacognitivity or not from this paper. A better question to ask will be: if the LLM doesn't have metacognitive capabilities, will this method work? Why or why not?

Questions

I asked some questions in Weakness section. Additionally, I'd suggest follow experiment: Instead of building a structured Skill Exemplar Repository, ask LLM to identify skill of current problem -> Recalled previous similar problem and answer with same skill -> Solve current problem -> evaluate solution. This process seem closer to human's problem solving process, and shows better evidence to support the argument "LLMs have metacognitive capabilities".

Rating

6

Confidence

4

Soundness

3

Presentation

3

Contribution

2

Limitations

Author mentioned the limitation of this method in paper as well: - It only assigns one skill to one math problem. Most complex math problem involves multiple skills to solve, thus it may not work well in that case.

Summary

This paper proposes a novel framework for extracting metacognitive knowledge from LLMs. Specifically, the training examples are firstly assigned with skill names. Then they are clustered with in semantic perspective. Finally they group the training examples as skill exemplars for the use of inference phase. Experimental results show that the proposed method can outperform CoT baselines.

Strengths

1. A novel framework is proposed to improve the math reasoning of LLMs by providing skill-based exemplars during the inference phase. 2. This paper proposes to leverage the metacognitive knowledge of LLMs by extracting the skill names of training examples, which is useful and interpretable to humans. 3. Experimental results on GSM8K and MATH show the effectiveness of proposed framework.

Weaknesses

1. The skill clusters appears to be useful; however, the definition of the number of clusters and the granularity of skill names lacks thorough analysis. Is it a tunable hyper-parameter when applying it to a new out-of-domain mathematical dataset? 2. Due to the variations in QA pairs across different datasets (such as answer format, question types, etc.), there are concerns regarding the generalization capability of the proposed method. For instance, if the Skill Exemplar Repository constructed on a training set is based on multiple-choice QA, what happens when the test set is in a non-multiple-choice format? Additionally, considering that the MATH dataset is more challenging than GSM8K, are there any experiments demonstrating the transfer from the MATH dataset to the GSM8K dataset?

Questions

See above questions

Rating

7

Confidence

4

Soundness

3

Presentation

3

Contribution

3

Limitations

n/a

Summary

This paper investigates whether large language models (LLMs) possess metacognitive knowledge, or an understanding of their own thinking and reasoning processes, particularly in the context of solving mathematical problems. Authors introduce a method to extract and utilize this metacognitive knowledge to enhance the reasoning abilities of LLMs. Specifically, A procedure is developed for LLMs to assign skill labels to math questions, followed by semantic clustering to group similar skills into broader categories, making the skill labels more interpretable to humans. Subsequently, a repository is created consisting of skill names and corresponding question/answer examples. This is used to provide in-context examples to guide problem-solving in LLMs. Experiments are conducted on GSM8K and MATH datasets.

Strengths

- The paper introduces a novel method for improving mathematical reasoning in LLMs by utilizing their metacognitive abilities, which is an original contribution to the field. - The authors back their claims with empirical evidence, demonstrating significant performance improvements in LLMs when using the proposed method. - The paper shows that the skills identified by a strong LLM can be transferred to other, less advanced models, enhancing their performance as well, which is a testament to the method's versatility. - The paper is well-organized, with a clear introduction, methodology, experimental setup, results, and discussion, making it easy to follow.

Weaknesses

- The paper's methodology assigns only one skill to each math question, which might be an oversimplification, as many problems require a combination of primary and secondary skills. What if assign multi-skills to each QA pair ? - The paper may not address the risk of overfitting to the training data when creating the skill exemplar repository. - Although the paper claims the methodology is domain-agnostic, concrete evidence of its effectiveness in non-mathematical domains is not provided. - Some other methods on solving MATH problem such as TORA [1] need to be discussed and compared. - While the paper provides details for reproducing the experiments, it does not provide open access to the code and data, which could limit the ability of other researchers to verify the results. [R1] ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving

Questions

- How does the model handle mathematical problems that require the application of multiple skills or a combination of primary and secondary skills? Does the single skill assignment per question potentially limit the model's ability to generalize? - If other methods such as TORA can benefit from the proposed approach? - How does the proposed method scale with the size of the dataset, and what is its computational complexity? Are there any optimizations that can be made to improve efficiency?

Rating

6

Confidence

3

Soundness

3

Presentation

3

Contribution

2

Limitations

Authors has adequately addressed the limitations.

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Follow-Up on Our Response and Request for Score Reconsideration

Dear Reviewer, We hope this message finds you well. We are writing to follow up on the new experimental results and explanations provided in response to your review. We genuinely appreciate the time and effort you’ve invested in reviewing our work. If our experiments and clarifications have adequately addressed your concerns, we kindly ask if you would consider revisiting your score. As August 13th marks the end of the discussion period, we would be grateful if you could share any additional questions or concerns you may have before then, so we can address them in a timely manner. Thank you once again for your thoughtful review.

Follow-Up on Our Response and Request for Score Reconsideration

Dear Reviewer, We hope this message finds you well. We are writing to follow up on the new experimental results and explanations provided in response to your review. We genuinely appreciate the time and effort you’ve invested in reviewing our work. If our experiments and clarifications have adequately addressed your concerns, we kindly ask if you would consider revisiting your score. As August 13th marks the end of the discussion period, we would be grateful if you could share any additional questions or concerns you may have before then, so we can address them in a timely manner. Thank you once again for your thoughtful review.