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Poster
SSRGD: Simple Stochastic Recursive Gradient Descent for Escaping Saddle Points
Zhize Li

Tue Dec 10 05:30 PM -- 07:30 PM (PST) @ East Exhibition Hall B + C #125
We analyze stochastic gradient algorithms for optimizing nonconvex problems. In particular, our goal is to find local minima (second-order stationary points) instead of just finding first-order stationary points which may be some bad unstable saddle points. We show that a simple perturbed version of stochastic recursive gradient descent algorithm (called SSRGD) can find an $(\epsilon,\delta)$-second-order stationary point with $\widetilde{O}(\sqrt{n}/\epsilon^2 + \sqrt{n}/\delta^4 + n/\delta^3)$ stochastic gradient complexity for nonconvex finite-sum problems. As a by-product, SSRGD finds an $\epsilon$-first-order stationary point with $O(n+\sqrt{n}/\epsilon^2)$ stochastic gradients. These results are almost optimal since Fang et al. [2018] provided a lower bound $\Omega(\sqrt{n}/\epsilon^2)$ for finding even just an $\epsilon$-first-order stationary point. We emphasize that SSRGD algorithm for finding second-order stationary points is as simple as for finding first-order stationary points just by adding a uniform perturbation sometimes, while all other algorithms for finding second-order stationary points with similar gradient complexity need to combine with a negative-curvature search subroutine (e.g., Neon2 [Allen-Zhu and Li, 2018]). Moreover, the simple SSRGD algorithm gets a simpler analysis. Besides, we also extend our results from nonconvex finite-sum problems to nonconvex online (expectation) problems, and prove the corresponding convergence results.

Author Information

Zhize Li (Tsinghua University, and KAUST)

Zhize Li is a Research Scientist at the King Abdullah University of Science and Technology (KAUST) since September 2020. He obtained his PhD degree in Computer Science from Tsinghua University in 2019 (Advisor: Prof. Jian Li). He was a postdoc at KAUST (Hosted by Prof. Peter Richtárik), a visiting scholar at Duke University (Hosted by Prof. Rong Ge), and a visiting scholar at Georgia Institute of Technology (Hosted by Prof. Guanghui (George) Lan).

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