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We consider a wide range of regularized stochastic minimization problems withtwo regularization terms, one of which is composed with a linear function. Thisoptimization model abstracts a number of important applications in artificialintelligence and machine learning, such as fused Lasso, fused logisticregression, and a class of graph-guided regularized minimization. Thecomputational challenges of this model are in two folds. On one hand, theclosed-form solution of the proximal mapping associated with the composedregularization term or the expected objective function is not available. On theother hand, the calculation of the full gradient of the expectation in theobjective is very expensive when the number of input data samples isconsiderably large. To address these issues, we propose a stochastic variant ofextra-gradient type methods, namely \textsf{Stochastic Primal-Dual ProximalExtraGradient descent (SPDPEG)}, and analyze its convergence property for bothconvex and strongly convex objectives. For general convex objectives, theuniformly average iterates generated by \textsf{SPDPEG} converge in expectationwith $O(1/\sqrt{t})$ rate. While for strongly convex objectives, the uniformlyand non-uniformly average iterates generated by \textsf{SPDPEG} converge with$O(\log(t)/t)$ and $O(1/t)$ rates, respectively. The order of the rate of theproposed algorithm is known to match the best convergence rate for first-orderstochastic algorithms. Experiments on fused logistic regression andgraph-guided regularized logistic regression problems show that the proposedalgorithm performs very efficiently and consistently outperforms othercompeting algorithms.

Stochastic Primal-Dual Proximal ExtraGradient Descent for Compositely Regularized Optimization