  We propose a novel stochastic smoothing accelerated gradient (SSAG) method
for general constrained nonsmooth convex composite optimization, and analyze
the convergence rates. The SSAG method allows various smoothing techniques, and
can deal with the nonsmooth term that is not easy to compute its proximal term,
or that does not own the linear max structure. To the best of our knowledge, it
is the first stochastic approximation type method with solid convergence result
to solve the convex composite optimization problem whose nonsmooth term is the
maximization of numerous nonlinear convex functions. We prove that the SSAG
method achieves the best-known complexity bounds in terms of the stochastic
first-order oracle ($\mathcal{SFO}$), using either diminishing smoothing
parameters or a fixed smoothing parameter. We give two applications of our
results to distributionally robust optimization problems. Numerical results on
the two applications demonstrate the effectiveness and efficiency of the
proposed SSAG method.
