A Functional Perspective on Learning Symmetric Functions with Neural Networks

TL;DR

This paper studies neural networks that learn symmetric functions over variable input sizes by treating them as functions over probability measures, providing theoretical bounds and empirical validation for their generalization capabilities.

Contribution

It introduces a measure-based framework for symmetric functions, establishing approximation and generalization bounds for shallow neural networks across varying input sizes.

Findings

Models can be learned efficiently with generalization guarantees.

Theoretical bounds depend on regularization choices like RKHS and variation norms.

Empirical results confirm cross-input size generalization.

Abstract

Symmetric functions, which take as input an unordered, fixed-size set, are known to be universally representable by neural networks that enforce permutation invariance. These architectures only give guarantees for fixed input sizes, yet in many practical applications, including point clouds and particle physics, a relevant notion of generalization should include varying the input size. In this work we treat symmetric functions (of any size) as functions over probability measures, and study the learning and representation of neural networks defined on measures. By focusing on shallow architectures, we establish approximation and generalization bounds under different choices of regularization (such as RKHS and variation norms), that capture a hierarchy of functional spaces with increasing degree of non-linear learning. The resulting models can be learned efficiently and enjoy generalization guarantees that extend across input sizes, as we verify empirically.