Distributionally Time-Varying Online Stochastic Optimization under Polyak-{\L}ojasiewicz Condition with Application in Conditional Value-at-Risk Statistical Learning

TL;DR

This paper develops a framework for online stochastic optimization under time-varying distributions satisfying the Polyak-Łojasiewicz condition, with applications to CVaR learning, providing regret bounds that account for distribution shifts and stochasticity.

Contribution

It introduces a novel analysis of online stochastic gradient methods under time-varying distributions using Wasserstein distance and applies this to improve CVaR learning regret bounds.

Findings

Established dynamic regret bounds considering distribution drift and stochastic bias.

Applied Wasserstein distance to handle non-absolute continuity and support variations.

Improved theoretical understanding of CVaR learning under time-varying distributions.

Abstract

In this work, we consider a sequence of stochastic optimization problems following a time-varying distribution via the lens of online optimization. Assuming that the loss function satisfies the Polyak-{\L}ojasiewicz condition, we apply online stochastic gradient descent and establish its dynamic regret bound that is composed of cumulative distribution drifts and cumulative gradient biases caused by stochasticity. The distribution metric we adopt here is Wasserstein distance, which is well-defined without the absolute continuity assumption or with a time-varying support set. We also establish a regret bound of online stochastic proximal gradient descent when the objective function is regularized. Moreover, we show that the above framework can be applied to the Conditional Value-at-Risk (CVaR) learning problem. Particularly, we improve an existing proof on the discovery of the PL condition of the CVaR problem, resulting in a regret bound of online stochastic gradient descent.