Learning Hierarchical Polynomials with Three-Layer Neural Networks

TL;DR

This paper proves that three-layer neural networks can efficiently learn hierarchical polynomial functions with polynomial sample complexity, outperforming kernel methods and previous neural network guarantees, especially for quadratic target functions.

Contribution

It establishes the first polynomial-time, polynomial-sample guarantees for learning hierarchical polynomials with three-layer neural networks, extending prior quadratic-specific results.

Findings

Neural networks learn hierarchical polynomials with $ ilde{O}(d^k)$ samples.

Three-layer networks outperform kernel methods for these functions.

Optimal sample complexity $ ilde{O}(d^2)$ achieved for quadratic polynomials.

Abstract

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form $h = g \circ p$ where $p : \mathbb{R}^d \rightarrow \mathbb{R}$ is a degree $k$ polynomial and $g: \mathbb{R} \rightarrow \mathbb{R}$ is a degree $q$ polynomial. This function class generalizes the single-index model, which corresponds to $k=1$, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree $k$ polynomials $p$, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target $h$ up to vanishing test error in $\widetilde{\mathcal{O}}(d^k)$ samples and polynomial time. This is a strict improvement over kernel methods, which require $\widetilde \Theta(d^{kq})$ samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of $p$ being a quadratic. When $p$ is indeed a quadratic, we achieve the information-theoretically optimal sample complexity $\widetilde{\mathcal{O}}(d^2)$, which is an improvement over prior work~\citep{nichani2023provable} requiring a sample size of $\widetilde\Theta(d^4)$. Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature $p$ with $\widetilde{\mathcal{O}}(d^k)$ samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.