Subgradient methods converge linearly on a convex function that grows sharply away from its solution set. In this work, we show that the same is true for sharp functions that are only weakly convex, provided that the subgradient methods are initialized within a fixed tube around the solution set. A variety of statistical and signal processing tasks come equipped with good initialization, and provably lead to formulations that are both weakly convex and sharp. Therefore, in such settings, subgradient methods can serve as inexpensive local search procedures. We illustrate the proposed techniques on phase retrieval and covariance estimation problems.

We study a new graph invariant, the sequence {sk} of the number of k‐edge subtrees of a graph. We compute the mean subtree size for several classes of graphs, concentrating on complete graphs, complete bipartite graphs, and theta graphs, in particular. We prove that the ratio of spanning trees to all subtrees in Kn approaches (1/e)(1/e)=0.692201⋯, and give a related formula for Kn,n. We also connect the number of subtrees of Kn that contain a given subtree to the hyperbinomial transform. For theta graphs, we find formulas for the mean subtree size (approximately 23n) and the mode (approximately 22n) of the unimodal sequence {sk}. The main tool is a subtree generating function.

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.

We revisit the foundations of gauge duality and demonstrate that it can be explained using a modern approach to duality based on a perturbation framework. We therefore put gauge duality and Fenchel-Rockafellar duality on equal footing, including explaining gauge dual variables as sensitivity measures, and showing how to recover primal solutions from those of the gauge dual. This vantage point allows a direct proof that optimal solutions of the Fenchel-Rockafellar dual of the gauge dual are precisely the primal solutions rescaled by the optimal value. We extend the gauge duality framework to the setting in which the functional components are general nonnegative convex functions, including problems with piecewise linear quadratic functions and constraints that arise from generalized linear models used in regression.