Momentum-Based Variance-Reduced Proximal Stochastic Gradient Method for Composite Nonconvex Stochastic Optimization

Stochastic gradient methods (SGMs) have been extensively used for solving stochastic problems or large-scale machine learning problems. Recent works employ various techniques to improve the convergence rate of SGMs for both convex and nonconvex cases. Most of them require a large number of samples in some or all iterations of the improved SGMs. In this paper, we propose a new SGM, named PStorm, for solving nonconvex nonsmooth stochastic problems. With a momentum-based variance reduction technique, PStorm can achieve the optimal complexity result $$O(\varepsilon ^{-3})$$ O ( ε - 3 ) to produce a stochastic $$\varepsilon $$ ε -stationary solution, if a mean-squared smoothness condition holds. Different from existing optimal methods, PStorm can achieve the $${O}(\varepsilon ^{-3})$$ O ( ε - 3 ) result by using only one or O(1) samples in every update. With this property, PStorm can be applied to online learning problems that favor real-time decisions based on one or O(1) new observations. In addition, for large-scale machine learning problems, PStorm can generalize better by small-batch training than other optimal methods that require large-batch training and the vanilla SGM, as we demonstrate on training a sparse fully-connected neural network and a sparse convolutional neural network.