Heavy-tailed Streaming Statistical Estimation

TL;DR

This paper introduces a new clipped stochastic gradient descent method for heavy-tailed streaming data, achieving exponential convergence guarantees with minimal batch size, applicable to mean estimation and linear regression.

Contribution

It presents an improved analysis of a clipped SGD algorithm under heavy-tailed noise, with theoretical guarantees and empirical validation for statistical estimation tasks.

Findings

Guarantees convergence with exponential concentration

Operates with O(1) batch size in streaming setting

Effective for mean estimation and linear regression

Abstract

We consider the task of heavy-tailed statistical estimation given streaming $p$-dimensional samples. This could also be viewed as stochastic optimization under heavy-tailed distributions, with an additional $O(p)$ space complexity constraint. We design a clipped stochastic gradient descent algorithm and provide an improved analysis, under a more nuanced condition on the noise of the stochastic gradients, which we show is critical when analyzing stochastic optimization problems arising from general statistical estimation problems. Our results guarantee convergence not just in expectation but with exponential concentration, and moreover does so using $O(1)$ batch size. We provide consequences of our results for mean estimation and linear regression. Finally, we provide empirical corroboration of our results and algorithms via synthetic experiments for mean estimation and linear regression.