Non-convex Finite-Sum Optimization Via SCSG Methods

Abstract

We develop a class of algorithms, as variants of the stochastically controlled stochastic gradient (SCSG) methods (Lei and Jordan, 2016), for the smooth non-convex finite-sum optimization problem. Assuming the smoothness of each component, the complexity of SCSG to reach a stationary point with $\mathbb{E}\Vert \nabla f(x){\Vert }^2\le \hat{I}\mu$ is $O\left(\min \{\hat{I}{\mu }^{-5/3},\hat{I}{\mu }^{-1}{n}^{2/3}\}\right)$, which strictly outperforms the stochastic gradient descent. Moreover, SCSG is never worse than the state-of-the-art methods based on variance reduction and it significantly outperforms them when the target accuracy is low. A similar acceleration is also achieved when the functions satisfy the Polyak-Lojasiewicz condition. Empirical experiments demonstrate that SCSG outperforms stochastic gradient methods on training multi-layers neural networks in terms of both training and validation loss.

Add Lemma B.1