Non-Convex Finite-Sum Optimization Via SCSG Methods

We develop a class of algorithms, as variants of the stochastically controlled stochastic gradient (SCSG) methods , for the smooth nonconvex finite-sum optimization problem. Only assuming the smoothness of each component, the complexity of SCSG to reach a stationary point with E∥∇f(x)∥2≤ϵE‖∇f(x)‖2≤ϵ is O(min{ϵ−5/3,ϵ−1n2/3})O(min{ϵ−5/3,ϵ−1n2/3}), 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.