  In this paper we introduce new methods for convex optimization problems with
inexact stochastic oracle. First method is an extension of the intermediate
gradient method proposed by Devolder, Glineur and Nesterov for problems with
inexact oracle. Our new method can be applied to the problems with composite
structure, stochastic inexact oracle and allows using non-Euclidean setup. We
prove estimates for mean rate of convergence and probabilities of large
deviations from this rate. Also we introduce two modifications of this method
for strongly convex problems. For the first modification we prove mean rate of
convergence estimates and for the second we prove estimates for large
deviations from the mean rate of convergence. All the rates give the complexity
estimates for proposed methods which up to multiplicative constant coincide
with lower complexity bound for the considered class of convex composite
optimization problems with stochastic inexact oracle.
