Sample Complexity of Stochastic Variance-Reduced Cubic Regularization for Nonconvex Optimization

The popular cubic regularization (CR) method converges with firstand second-order optimality guarantee for nonconvex optimization, but encounters a high sample complexity issue for solving large-scale problems. Various sub-sampling variants of CR have been proposed to improve the sample complexity. In this paper, we propose a stochastic variance-reduced cubic-regularized (SVRC) Newton’s method under both sampling with and without replacement schemes. We characterize the per-iteration sample complexity bounds which guarantee the same rate of convergence as that of CR for nonconvex optimization. Furthermore, our method achieves a total Hessian sample complexity of O(N −3/2) and O(N −3/2) respectively under sampling without and with replacement, which improve that of CR as well as other sub-sampling variant methods via the variance reduction scheme. Our result also suggests that sampling without replacement yields lower sample complexity than that of sampling with replacement. We further compare the practical performance of SVRC with other cubic regularization methods via experiments.