  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.
