Sharp Analysis of Stochastic Optimization under Global Kurdyka-Łojasiewicz Inequality

We study the complexity of finding the global solution to stochastic nonconvex optimization when the objective function satisfies global Kurdyka-Lojasiewicz (KL) inequality and the queries from stochastic gradient oracles satisfy mild expected smoothness assumption. We first introduce a general framework to analyze Stochastic Gradient Descent (SGD) and its associated nonlinear dynamics under the setting. As a byproduct of our analysis, we obtain a sample complexity of $\mathcal{O}(ε^{-(4-α)/α})$ for SGD when the objective satisfies the so called $α$-PL condition, where $α$ is the degree of gradient domination. Furthermore, we show that a modified SGD with variance reduction and restarting (PAGER) achieves an improved sample complexity of $\mathcal{O}(ε^{-2/α})$ when the objective satisfies the average smoothness assumption. This leads to the first optimal algorithm for the important case of $α=1$ which appears in applications such as policy optimization in reinforcement learning.