Robust stochastic optimization with the proximal point method

Standard results in stochastic convex optimization bound the number of samples that an algorithm needs to generate a point with small function value in expectation. In this work, we show that a wide class of such algorithms on strongly convex problems can be augmented with high confidence bounds at an overhead cost that is only logarithmic in the confidence level and polylogarithmic in the condition number. We discuss consequences both for streaming and offline algorithms.