  Stochastic gradient algorithms estimate the gradient based on only one or a
few samples and enjoy low computational cost per iteration. They have been
widely used in large-scale optimization problems. However, stochastic gradient
algorithms are usually slow to converge and achieve sub-linear convergence
rates, due to the inherent variance in the gradient computation. To accelerate
the convergence, some variance-reduced stochastic gradient algorithms, e.g.,
proximal stochastic variance-reduced gradient (Prox-SVRG) algorithm, have
recently been proposed to solve strongly convex problems. Under the strongly
convex condition, these variance-reduced stochastic gradient algorithms achieve
a linear convergence rate. However, many machine learning problems are convex
but not strongly convex. In this paper, we introduce Prox-SVRG and its
projected variant called Variance-Reduced Projected Stochastic Gradient (VRPSG)
to solve a class of non-strongly convex optimization problems widely used in
machine learning. As the main technical contribution of this paper, we show
that both VRPSG and Prox-SVRG achieve a linear convergence rate without strong
convexity. A key ingredient in our proof is a Semi-Strongly Convex (SSC)
inequality which is the first to be rigorously proved for a class of
non-strongly convex problems in both constrained and regularized settings.
Moreover, the SSC inequality is independent of algorithms and may be applied to
analyze other stochastic gradient algorithms besides VRPSG and Prox-SVRG, which
may be of independent interest. To the best of our knowledge, this is the first
work that establishes the linear convergence rate for the variance-reduced
stochastic gradient algorithms on solving both constrained and regularized
problems without strong convexity.
