## Sharp Analysis of Stochastic Optimization under Global Kurdyka-Lojasiewicz Inequality

### Ilyas Fatkhullin · Jalal Etesami · Niao He · Negar Kiyavash

##### Hall J #841

Keywords: [ nonconvex optimization ] [ variance reduction ] [ Stochastic Optimization ] [ Kurdyka-Lojasiewicz condition ] [ first order method ]

[ Abstract ]
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Wed 30 Nov 2 p.m. PST — 4 p.m. PST

Abstract: 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.

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