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Stochastic Block Mirror Descent Methods for Nonsmooth and Stochastic Optimization
In this paper, we present a new stochastic algorithm, namely the stochastic
block mirror descent (SBMD) method for solving large-scale nonsmooth and
stochastic optimization problems. The basic idea of this algorithm is to
incorporate the block-coordinate decomposition and an incremental block
averaging scheme into the classic (stochastic) mirror-descent method, in order
to significantly reduce the cost per iteration of the latter algorithm. We
establish the rate of convergence of the SBMD method along with its associated
large-deviation results for solving general nonsmooth and stochastic
optimization problems. We also introduce different variants of this method and
establish their rate of convergence for solving strongly convex, smooth, and
composite optimization problems, as well as certain nonconvex optimization
problems. To the best of our knowledge, all these developments related to the
SBMD methods are new in the stochastic optimization literature. Moreover, some
of our results also seem to be new for block coordinate descent methods for
deterministic optimization
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