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A Stochastic Gradient Method with Linear Convergence Rate for a Class of Non-smooth Non-strongly Convex Optimizations

In this paper, we show that a (stochastic) gradient decent method with multiple restarting, named Restarted (S)GD, can achieve a linear convergence rate for a class of non-smooth and non-strongly convex optimization problems where the epigraph of the objective function is a polyhedron. Its applications in machine learning include minimizing `1 constrained or regularized piecewise linear loss. To the best of our knowledge, this is the first result on the linear convergence rate of a stochastic gradient method for non-smooth and non-strongly convex optimization.