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In this paper, we present a new stochastic algorithm, namely the stochasticblock mirror descent (SBMD) method for solving large-scale nonsmooth andstochastic optimization problems. The basic idea of this algorithm is toincorporate the block-coordinate decomposition and an incremental blockaveraging scheme into the classic (stochastic) mirror-descent method, in orderto significantly reduce the cost per iteration of the latter algorithm. Weestablish the rate of convergence of the SBMD method along with its associatedlarge-deviation results for solving general nonsmooth and stochasticoptimization problems. We also introduce different variants of this method andestablish their rate of convergence for solving strongly convex, smooth, andcomposite optimization problems, as well as certain nonconvex optimizationproblems. To the best of our knowledge, all these developments related to theSBMD methods are new in the stochastic optimization literature. Moreover, someof our results also seem to be new for block coordinate descent methods fordeterministic optimization.

Stochastic Block Mirror Descent Methods for Nonsmooth and Stochastic Optimization