  In this paper, we propose a stochastic Primal-Dual Hybrid Gradient (PDHG)
approach for solving a wide spectrum of regularized stochastic minimization
problems, where the regularization term is composite with a linear function. It
has been recognized that solving this kind of problem is challenging since the
closed-form solution of the proximal mapping associated with the regularization
term is not available due to the imposed linear composition, and the
per-iteration cost of computing the full gradient of the expected objective
function is extremely high when the number of input data samples is
considerably large.
  Our new approach overcomes these issues by exploring the special structure of
the regularization term and sampling a few data points at each iteration.
Rather than analyzing the convergence in expectation, we provide the detailed
iteration complexity analysis for the cases of both uniformly and non-uniformly
averaged iterates with high probability. This strongly supports the good
practical performance of the proposed approach. Numerical experiments
demonstrate that the efficiency of stochastic PDHG, which outperforms other
competing algorithms, as expected by the high-probability convergence analysis.
