Near-Optimal Streaming Heavy-Tailed Statistical Estimation with Clipped SGD

TL;DR

This paper demonstrates that Clipped-SGD achieves near-optimal statistical rates for high-dimensional heavy-tailed data in streaming settings, extending its effectiveness beyond strongly convex functions.

Contribution

The authors establish near-optimal convergence rates for Clipped-SGD in heavy-tailed streaming data, introducing a new iterative martingale concentration technique.

Findings

Clipped-SGD attains near-optimal sub-Gaussian rates with finite second moment of gradients.

The error bound improves previous rates by reducing dependence on $rac{1}{\delta}$.

Results extend to smooth and Lipschitz convex objectives.

Abstract

We consider the problem of high-dimensional heavy-tailed statistical estimation in the streaming setting, which is much harder than the traditional batch setting due to memory constraints. We cast this problem as stochastic convex optimization with heavy tailed stochastic gradients, and prove that the widely used Clipped-SGD algorithm attains near-optimal sub-Gaussian statistical rates whenever the second moment of the stochastic gradient noise is finite. More precisely, with $T$ samples, we show that Clipped-SGD, for smooth and strongly convex objectives, achieves an error of $\sqrt{\frac{\mathsf{Tr}(\Sigma)+\sqrt{\mathsf{Tr}(\Sigma)\|\Sigma\|_2}\log(\frac{\log(T)}{\delta})}{T}}$ with probability $1-\delta$, where $\Sigma$ is the covariance of the clipped gradient. Note that the fluctuations (depending on $\frac{1}{\delta}$) are of lower order than the term $\mathsf{Tr}(\Sigma)$. This improves upon the current best rate of $\sqrt{\frac{\mathsf{Tr}(\Sigma)\log(\frac{1}{\delta})}{T}}$ for Clipped-SGD, known only for smooth and strongly convex objectives. Our results also extend to smooth convex and lipschitz convex objectives. Key to our result is a novel iterative refinement strategy for martingale concentration, improving upon the PAC-Bayes approach of Catoni and Giulini.