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We study the complexity of finding the global solution to stochasticnonconvex optimization when the objective function satisfies globalKurdyka-Lojasiewicz (KL) inequality and the queries from stochastic gradientoracles satisfy mild expected smoothness assumption. We first introduce ageneral framework to analyze Stochastic Gradient Descent (SGD) and itsassociated nonlinear dynamics under the setting. As a byproduct of ouranalysis, we obtain a sample complexity of$\mathcal{O}(\epsilon^{-(4-\alpha)/\alpha})$ for SGD when the objectivesatisfies the so called $\alpha$-PL condition, where $\alpha$ is the degree ofgradient domination. Furthermore, we show that a modified SGD with variancereduction and restarting (PAGER) achieves an improved sample complexity of$\mathcal{O}(\epsilon^{-2/\alpha})$ when the objective satisfies the averagesmoothness assumption. This leads to the first optimal algorithm for theimportant case of $\alpha=1$ which appears in applications such as policyoptimization in reinforcement learning.

Sharp Analysis of Stochastic Optimization under Global Kurdyka-{\L}ojasiewicz Inequality