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

Distributed Stochastic Optimization of the Regularized Risk

Many machine learning algorithms minimize a regularized risk, and stochastic optimization is widely used for this task. When working with massive data, it is desirable to perform stochastic optimization in parallel. Unfortunately, many existing stochastic optimization algorithms cannot be parallelized efficiently. In this paper we show that one can rewrite the regularized risk minimization problem as an equivalent saddle-point problem, and propose an efficient distributed stochastic optimization (DSO) algorithm. We prove the algorithm's rate of convergence; remarkably, our analysis shows that the algorithm scales almost linearly with the number of processors. We also verify with empirical evaluations that the proposed algorithm is competitive with other parallel, general purpose stochastic and batch optimization algorithms for regularized risk minimization