Clipped Gradient Methods for Nonsmooth Convex Optimization under Heavy-Tailed Noise: A Refined Analysis

TL;DR

This paper refines the analysis of Clipped SGD for nonsmooth convex optimization under heavy-tailed noise, achieving faster convergence rates and establishing optimality through new lower bounds.

Contribution

It provides improved convergence rates for Clipped SGD using a refined analysis and introduces the concept of generalized effective dimension.

Findings

Faster convergence rates under heavy-tailed noise.

New lower bounds matching the upper bounds.

Enhanced analysis utilizing Freedman's inequality.

Abstract

Optimization under heavy-tailed noise has become popular recently, since it better fits many modern machine learning tasks, as captured by empirical observations. Concretely, instead of a finite second moment on gradient noise, a bounded ${\frak p}$-th moment where ${\frak p}\in(1,2]$ has been recognized to be more realistic (say being upper bounded by $\sigma_{\frak l}^{\frak p}$ for some $\sigma_{\frak l}\ge0$). A simple yet effective operation, gradient clipping, is known to handle this new challenge successfully. Specifically, Clipped Stochastic Gradient Descent (Clipped SGD) guarantees a high-probability rate ${\cal O}(\sigma_{\frak l}\ln(1/\delta)T^{1/{\frak p}-1})$ (resp. ${\cal O}(\sigma_{\frak l}^2\ln^2(1/\delta)T^{2/{\frak p}-2})$) for nonsmooth convex (resp. strongly convex) problems, where $\delta\in(0,1]$ is the failure probability and $T\in\mathbb{N}$ is the time horizon. In this work, we provide a refined analysis for Clipped SGD and offer two rates, ${\cal O}(\sigma_{\frak l}d_{\rm eff}^{-1/2{\frak p}}\ln^{1-1/{\frak p}}(1/\delta)T^{1/{\frak p}-1})$ and ${\cal O}(\sigma_{\frak l}^2d_{\rm eff}^{-1/{\frak p}}\ln^{2-2/{\frak p}}(1/\delta)T^{2/{\frak p}-2})$, faster than the aforementioned best results, where $d_{\rm eff}\ge1$ is a quantity we call the $\textit{generalized effective dimension}$. Our analysis improves upon the existing approach on two sides: better utilization of Freedman's inequality and finer bounds for clipping error under heavy-tailed noise. In addition, we extend the refined analysis to convergence in expectation and obtain new rates that break the known lower bounds. Lastly, to complement the study, we establish new lower bounds for both high-probability and in-expectation convergence. Notably, the in-expectation lower bounds match our new upper bounds, indicating the optimality of our refined analysis for convergence in expectation.