Faster Perturbed Stochastic Gradient Methods for Finding Local Minima

Escaping from saddle points and finding local minimum is a central problem in nonconvex optimization. Perturbed gradient methods are perhaps the simplest approach for this problem. However, to find $(ε, \sqrtε)$-approximate local minima, the existing best stochastic gradient complexity for this type of algorithms is $\tilde O(ε^{-3.5})$, which is not optimal. In this paper, we propose LENA (Last stEp shriNkAge), a faster perturbed stochastic gradient framework for finding local minima. We show that LENA with stochastic gradient estimators such as SARAH/SPIDER and STORM can find $(ε, ε_{H})$-approximate local minima within $\tilde O(ε^{-3} + ε_{H}^{-6})$ stochastic gradient evaluations (or $\tilde O(ε^{-3})$ when $ε_H = \sqrtε$). The core idea of our framework is a step-size shrinkage scheme to control the average movement of the iterates, which leads to faster convergence to the local minima.

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