Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models

TL;DR

This paper demonstrates that smoothing the loss landscape enables stochastic gradient descent to learn single index models with optimal sample complexity, matching the lower bounds and improving understanding of SGD's efficiency.

Contribution

It shows that smoothing the loss landscape allows SGD to achieve optimal sample complexity for learning single index models, closing previous theoretical gaps.

Findings

SGD on smoothed loss learns with $n ceil d^{k^*/2}$ samples

Connects smoothing techniques to tensor PCA analysis

Highlights implicit regularization effects of minibatch SGD

Abstract

We focus on the task of learning a single index model $\sigma(w^\star \cdot x)$ with respect to the isotropic Gaussian distribution in $d$ dimensions. Prior work has shown that the sample complexity of learning $w^\star$ is governed by the information exponent $k^\star$ of the link function $\sigma$, which is defined as the index of the first nonzero Hermite coefficient of $\sigma$. Ben Arous et al. (2021) showed that $n \gtrsim d^{k^\star-1}$ samples suffice for learning $w^\star$ and that this is tight for online SGD. However, the CSQ lower bound for gradient based methods only shows that $n \gtrsim d^{k^\star/2}$ samples are necessary. In this work, we close the gap between the upper and lower bounds by showing that online SGD on a smoothed loss learns $w^\star$ with $n \gtrsim d^{k^\star/2}$ samples. We also draw connections to statistical analyses of tensor PCA and to the implicit regularization effects of minibatch SGD on empirical losses.