Symmetric Single Index Learning

TL;DR

This paper extends the theoretical understanding of single index models to symmetric neural networks, showing gradient flow can recover hidden directions under certain conditions, with a new notion of information exponent.

Contribution

It introduces a framework for analyzing single index learning in symmetric neural networks, including conditions for successful recovery and a new information exponent concept.

Findings

Gradient flow recovers the hidden direction in symmetric neural networks.

A new information exponent controls the learning efficiency.

Under certain assumptions, the model achieves polynomial sample complexity.

Abstract

Few neural architectures lend themselves to provable learning with gradient based methods. One popular model is the single-index model, in which labels are produced by composing an unknown linear projection with a possibly unknown scalar link function. Learning this model with SGD is relatively well-understood, whereby the so-called information exponent of the link function governs a polynomial sample complexity rate. However, extending this analysis to deeper or more complicated architectures remains challenging. In this work, we consider single index learning in the setting of symmetric neural networks. Under analytic assumptions on the activation and maximum degree assumptions on the link function, we prove that gradient flow recovers the hidden planted direction, represented as a finitely supported vector in the feature space of power sum polynomials. We characterize a notion of information exponent adapted to our setting that controls the efficiency of learning.