  This paper considers a class of convex constrained nonsmooth convex
stochastic composite optimization problems whose objective function is given by
the summation of a differentiable convex component, together with a general
nonsmooth but convex component. The nonsmooth component is not required to have
easily obtainable proximal operator, or have the max structure that the
smoothing technique in [Nesterov, 2005] can be used. In order to solve such
type problems, we propose an adaptive smoothing mini-batch stochastic
accelerated gradient (AdaSMSAG) method, which combines the stochastic
approximation method, the Nesterov's accelerated gradient method, and the
smoothing methods that allow general smoothing approximations. Convergence of
the method is established. Moreover, the order of the worst-case iteration
complexity is better than that of the state-of-the-art stochastic approximation
methods. Numerical results are provided to illustrate the efficiency of the
proposed AdaSMSAG method for a risk management in portfolio optimization and a
family of Wasserstein distributionally robust support vector machine problems
with real data.
