Stochastic ISTA/FISTA Adaptive Step Search Algorithms for Convex Composite Optimization

We develop and analyze stochastic variants of ISTA and a full backtracking FISTA algorithms [Beck and Teboulle, 2009, Scheinberg et al., 2014] for composite optimization without the assumption that stochastic gradient is an unbiased estimator. This work extends analysis of inexact fixed step ISTA/FISTA in [Schmidt et al., 2011] to the case of stochastic gradient estimates and adaptive step-size parameter chosen by backtracking. It also extends the framework for analyzing stochastic line-search method in [Cartis and Scheinberg, 2018] to the proximal gradient framework as well as to the accelerated first order methods.