Necoara (Bucharest Politehnica): Conditions for Linear Convergence of (Stochastic) First Order Methods

For convex optimization problems deterministic first order methods have linear convergence provided that the objective function is smooth (Lipschitz continuous gradient) and strongly convex. Moreover, under the same conditions - smoothness and strong convexity -  stochastic first order methods have sublinear convergence rates. However, in many applications (machine learning, statistics, control, signal processing) the smoothness/strong convexity conditions do not hold; but the objective function still has a special structure (e.g. composition of a strongly convex function with a linear map). In this talk we replace the smoothness/strong convexity assumptions with several other conditions, that are less conservative, for which we prove that several (stochastic) first order methods are converging linearly.  We also provide necessary conditions for linear convergence of (stochastic) gradient method.  Finally, we provide examples of several functional classes satisfying our new conditions and discuss several applications of these results (Lasso problem, linear systems, linear programming, convex feasibility, etc).
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