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In this paper, we propose a stochastic Primal-Dual Hybrid Gradient (PDHG)approach for solving a wide spectrum of regularized stochastic minimizationproblems, where the regularization term is composite with a linear function. Ithas been recognized that solving this kind of problem is challenging since theclosed-form solution of the proximal mapping associated with the regularizationterm is not available due to the imposed linear composition, and theper-iteration cost of computing the full gradient of the expected objectivefunction is extremely high when the number of input data samples isconsiderably large.  Our new approach overcomes these issues by exploring the special structure ofthe regularization term and sampling a few data points at each iteration.Rather than analyzing the convergence in expectation, we provide the detailediteration complexity analysis for the cases of both uniformly and non-uniformlyaveraged iterates with high probability. This strongly supports the goodpractical performance of the proposed approach. Numerical experimentsdemonstrate that the efficiency of stochastic PDHG, which outperforms othercompeting algorithms, as expected by the high-probability convergence analysis.

On the Iteration Complexity Analysis of Stochastic Primal-Dual Hybrid Gradient Approach with High Probability