Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track
Ilyas Fatkhullin, Jalal Etesami, Niao He, Negar Kiyavash
We study the complexity of finding the global solution to stochastic nonconvex optimization when the objective function satisfies global Kurdyka-{\L}ojasiewicz (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 $\mathcal{O}(\epsilon^{-(4-\alpha)/\alpha})$ for SGD when the objective satisfies the so called $\alpha$-P{\L} condition, where $\alpha$ 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 $\mathcal{O}(\epsilon^{-2/\alpha})$ when the objective satisfies the average smoothness assumption. This leads to the first optimal algorithm for the important case of $\alpha=1$ which appears in applications such as policy optimization in reinforcement learning.