SSRGD: Simple Stochastic Recursive Gradient Descent for Escaping Saddle Points
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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 (ϵ,δ)-second-order stationary point with O(√(n)/ϵ^2 + √(n)/δ^4 + n/δ^3) stochastic gradient complexity for nonconvex finite-sum problems. As a by-product, SSRGD finds an ϵ-first-order stationary point with O(n+√(n)/ϵ^2) stochastic gradients. These results are almost optimal since Fang et al. [2018] provided a lower bound Ω(√(n)/ϵ^2) for finding even just an ϵ-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.
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