Inexact SARAH Algorithm for Stochastic Optimization

We develop and analyze a variant of variance reducing stochastic gradient algorithm, known as SARAH, which does not require computation of the exact gradient. Thus this new method can be applied to general expectation minimization problems rather than only finite sum problems. While the original SARAH algorithm, as well as its predecessor, SVRG, require an exact gradient computation on each outer iteration, the inexact variant of SARAH (iSARAH), which we develop here, requires only stochastic gradient computed on a mini-batch of sufficient size. The proposed method combines variance reduction via sample size selection and iterative stochastic gradient updates. We analyze the convergence rate of the algorithms for strongly convex, convex, and nonconvex cases with appropriate mini-batch size selected for each case. We show that with an additional, reasonable, assumption iSARAH achieves the best known complexity among stochastic methods in the case of general convex case stochastic value functions.