Stochastic model-based minimization under high-order growth

07/01/2018

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by   Damek Davis, et al.

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Given a nonsmooth, nonconvex minimization problem, we consider algorithms that iteratively sample and minimize stochastic convex models of the objective function. Assuming that the one-sided approximation quality and the variation of the models is controlled by a Bregman divergence, we show that the scheme drives a natural stationarity measure to zero at the rate O(k^-1/4). Under additional convexity and relative strong convexity assumptions, the function values converge to the minimum at the rate of O(k^-1/2) and O(k^-1), respectively. We discuss consequences for stochastic proximal point, mirror descent, regularized Gauss-Newton, and saddle point algorithms.