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Machine learning and information theory concepts towards an AI Mathematician (2403.04571v1)

Published 7 Mar 2024 in cs.AI

Abstract: The current state-of-the-art in artificial intelligence is impressive, especially in terms of mastery of language, but not so much in terms of mathematical reasoning. What could be missing? Can we learn something useful about that gap from how the brains of mathematicians go about their craft? This essay builds on the idea that current deep learning mostly succeeds at system 1 abilities -- which correspond to our intuition and habitual behaviors -- but still lacks something important regarding system 2 abilities -- which include reasoning and robust uncertainty estimation. It takes an information-theoretical posture to ask questions about what constitutes an interesting mathematical statement, which could guide future work in crafting an AI mathematician. The focus is not on proving a given theorem but on discovering new and interesting conjectures. The central hypothesis is that a desirable body of theorems better summarizes the set of all provable statements, for example by having a small description length while at the same time being close (in terms of number of derivation steps) to many provable statements.

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Citations (8)
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Summary

  • The paper introduces a framework where AI generates and verifies conjectures using an information-theoretical view to compress mathematical knowledge.
  • It leverages cognitive principles, integrating working memory constraints and heuristic reasoning to bridge intuitive and deliberate thought processes in machine learning.
  • The approach employs active learning and reinforcement strategies to explore the space of provable statements, potentially guiding the discovery of new theorems.

Toward an AI Mathematician: Exploring Mathematical Discovery through Machine Learning and Information Theory

Introduction to the Challenge

The quest for an AI capable of human-level mathematical reasoning and theorem discovery is a fundamental challenge in AI research. As we explore the capabilities of generative AI and LLMs, the contrast between their linguistic prowess and their relatively underdeveloped capacity for mathematical reasoning becomes stark. This paper by Yoshua Bengio and Nikolay Malkin takes a novel approach to bridging this gap, proposing a framework for an AI mathematician that focuses on the generation and verification of mathematical conjectures rather than on proving pre-defined theorems. Central to their thesis is an information-theoretical view of mathematical statements, aiming to create an AI that can identify and explore theorems that offer the best compression of mathematical knowledge.

Cognition and AI: Bridging Two Systems of Thought

The distinction between human "System 1" abilities (intuition and habitual behaviors) and "System 2" abilities (deliberate reasoning and uncertainty estimation) sets the stage for understanding the current limitations of AI in mimicking mathematical thought processes. AI, particularly in the form of deep neural networks and LLMs, has shown impressive System 1 capabilities but lags significantly in System 2 reasoning, crucial for mathematical inquiry. The paper advocates for incorporating cognitive principles, such as working memory constraints and the generation of compositional discrete thoughts, as a pathway towards imbuing AI with human-like mathematical reasoning capabilities.

The Role of Compression in Mathematical Discovery

At the heart of the proposed approach is the principle of compression, a concept well-established in learning theory. This parallels the process of mathematical theorem discovery, wherein the "usefulness" of a theorem is associated with its ability to simplify or compress the space of provable mathematical statements. By adopting an information-theoretical perspective, the authors hypothesize that an optimal set of theorems would serve to efficiently summarize and compress all provable statements, potentially guiding the development of an AI mathematician towards the discovery of new, interesting conjectures.

Navigating the Space of Provable Statements

The exploration of the space of mathematical statements through generative models introduces a formalism akin to reinforcement learning, where the "actions" entail derivation steps leading to new conjectures. This approach underscores the adaptability of AI in navigating the vast and complex territory of mathematical knowledge, mirroring the conjecture-proof cycle typical of human mathematical activity. The authors put forth a compelling argument that the exploration process itself, guided by an information-theoretic objective, can yield insights into the intrinsic interestingness of mathematical statements.

Active Learning and Conjecture Generation

The invocation of active learning principles and goal-conditioned exploration offers a dynamic strategy for enhancing the generative capabilities of an AI mathematician. By learning to prioritize conjectures based on their novelty or surprising nature, AI can emulate the human penchant for pursuing theorems that expand or challenge the existing mathematical canon. This section elucidates the potential for leveraging uncertainty and epistemic curiosity as drivers for mathematical innovation.

Architectural Considerations and Future Directions

Delving into the technical architecture required for realizing such an AI mathematician, the paper discusses the integration of proof tactics, lemma generation, and hierarchical reinforcement learning strategies as essential components. These mechanisms mirror human heuristic and abstraction processes, pointing towards an AI system capable of autonomous theorem discovery and proof generation. Importantly, the paper acknowledges the iterative nature of this research journey, inviting further exploration into the balance between pre-training on human mathematical literature and unsupervised exploration within the mathematical space.

Concluding Thoughts

Yoshua Bengio and Nikolay Malkin's paper lays a foundational framework for approaching the grand challenge of developing an AI mathematician. By intertwining concepts from machine learning, information theory, and cognitive science, the authors chart a multi-faceted research agenda aimed at uncovering the mechanisms of mathematical discovery and reasoning. As AI continues to evolve, the vision of a machine not just solving but also proposing meaningful mathematical conjectures moves closer to reality, promising to redefine our understanding of intelligence and creativity in the digital field.

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