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In this work, we consider a sequence of stochastic optimization problemsfollowing 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 regretbound that is composed of cumulative distribution drifts and cumulativegradient biases caused by stochasticity. The distribution metric we adopt hereis Wasserstein distance, which is well-defined without the absolute continuityassumption or with a time-varying support set. We also establish a regret boundof online stochastic proximal gradient descent when the objective function isregularized. Moreover, we show that the above framework can be applied to theConditional Value-at-Risk (CVaR) learning problem. Particularly, we improve anexisting proof on the discovery of the PL condition of the CVaR problem,resulting in a regret bound of online stochastic gradient descent.

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