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The stochastic gradient (SG) method can minimize an objective functioncomposed of a large number of differentiable functions, or solve a stochasticoptimization problem, to a moderate accuracy. The block coordinatedescent/update (BCD) method, on the other hand, handles problems with multipleblocks of variables by updating them one at a time; when the blocks ofvariables are easier to update individually than together, BCD has a lowerper-iteration cost. This paper introduces a method that combines the featuresof SG and BCD for problems with many components in the objective and withmultiple (blocks of) variables.  Specifically, a block stochastic gradient (BSG) method is proposed forsolving both convex and nonconvex programs. At each iteration, BSG approximatesthe gradient of the differentiable part of the objective by randomly sampling asmall set of data or sampling a few functions from the sum term in theobjective, and then, using those samples, it updates all the blocks ofvariables in either a deterministic or a randomly shuffled order. Itsconvergence for both convex and nonconvex cases are established in differentsenses. In the convex case, the proposed method has the same order ofconvergence rate as the SG method. In the nonconvex case, its convergence isestablished in terms of the expected violation of a first-order optimalitycondition. The proposed method was numerically tested on problems includingstochastic least squares and logistic regression, which are convex, as well aslow-rank tensor recovery and bilinear logistic regression, which are nonconvex.

Block stochastic gradient iteration for convex and nonconvex optimization