  In this paper, we propose a unified view of gradient-based algorithms for
stochastic convex composite optimization by extending the concept of estimate
sequence introduced by Nesterov. More precisely, we interpret a large class of
stochastic optimization methods as procedures that iteratively minimize a
surrogate of the objective, which covers the stochastic gradient descent method
and variants of the incremental approaches SAGA, SVRG, and MISO/Finito/SDCA.
This point of view has several advantages: (i) we provide a simple generic
proof of convergence for all of the aforementioned methods; (ii) we naturally
obtain new algorithms with the same guarantees; (iii) we derive generic
strategies to make these algorithms robust to stochastic noise, which is useful
when data is corrupted by small random perturbations. Finally, we propose a new
accelerated stochastic gradient descent algorithm and an accelerated SVRG
algorithm with optimal complexity that is robust to stochastic noise.
