Can Neural Networks Learn Small Algebraic Worlds? An Investigation Into the Group-theoretic Structures Learned By Narrow Models Trained To Predict Group Operations

TL;DR

This paper investigates whether narrow neural networks trained on group operations can learn and represent fundamental algebraic structures, revealing some capabilities and limitations in capturing abstract mathematical concepts.

Contribution

It introduces a suite of tests to evaluate the extent to which neural networks learn group-theoretic properties and demonstrates that models can capture some algebraic notions like subgroups and commutativity.

Findings

Models can distinguish elements of certain subgroups without explicit labels.

Hints that models capture the commutativity of modular arithmetic.

Difficulty in extracting the concept of the identity element.

Abstract

While a real-world research program in mathematics may be guided by a motivating question, the process of mathematical discovery is typically open-ended. Ideally, exploration needed to answer the original question will reveal new structures, patterns, and insights that are valuable in their own right. This contrasts with the exam-style paradigm in which the machine learning community typically applies AI to math. To maximize progress in mathematics using AI, we will need to go beyond simple question answering. With this in mind, we explore the extent to which narrow models trained to solve a fixed mathematical task learn broader mathematical structure that can be extracted by a researcher or other AI system. As a basic test case for this, we use the task of training a neural network to predict a group operation (for example, performing modular arithmetic or composition of permutations). We describe a suite of tests designed to assess whether the model captures significant group-theoretic notions such as the identity element, commutativity, or subgroups. Through extensive experimentation we find evidence that models learn representations capable of capturing abstract algebraic properties. For example, we find hints that models capture the commutativity of modular arithmetic. We are also able to train linear classifiers that reliably distinguish between elements of certain subgroups (even though no labels for these subgroups are included in the data). On the other hand, we are unable to extract notions such as the concept of the identity element. Together, our results suggest that in some cases the representations of even small neural networks can be used to distill interesting abstract structure from new mathematical objects.