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

• URL: http://arxiv.org/abs/2305.10633v1
• Date: Thu, 18 May 2023 01:10:11 GMT
• Title: Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models
• Authors: Alex Damian, Eshaan Nichani, Rong Ge, Jason D. Lee
• Abstract summary: We focus on the task of learning a single index model $sigma(wstar cdot x)$ with respect to the isotropic Gaussian distribution in $d$ dimensions. We show that online SGD on a smoothed loss learns $wstar$ with $n gtrsim dkstar/2$ samples.
• Score: 43.83997656986799
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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