A Stochastic Nesterov’s Smoothing Accelerated Method for General Nonsmooth Constrained Stochastic Composite Convex Optimization

We propose a novel stochastic Nesterov’s smoothing accelerated method for general nonsmooth, constrained, stochastic composite convex optimization, the nonsmooth component of which may be not easy to compute its proximal operator. The proposed method combines Nesterov’s smoothing accelerated method (Nesterov in Math Program 103(1):127–152, 2005) for deterministic problems and stochastic approximation for stochastic problems, which allows three variants: single sample and two different mini-batch sizes per iteration, respectively. We prove that all the three variants achieve the best-known complexity bounds in terms of stochastic oracle. Numerical results on a robust linear regression problem, as well as a support vector machine problem show that the proposed method compares favorably with other state-of-the-art first-order methods, and the variants with mini-batch sizes outperform the variant with single sample.