Subgradient methods for sharp weakly convex functions

TL;DR

This paper extends the linear convergence guarantees of subgradient methods from convex functions to sharp, weakly convex functions, given proper initialization, with applications in phase retrieval and covariance estimation.

Contribution

It demonstrates that subgradient methods converge linearly for sharp, weakly convex functions when initialized within a certain region, broadening their applicability.

Findings

Subgradient methods achieve linear convergence on weakly convex, sharp functions with proper initialization.

Applications include phase retrieval and covariance estimation tasks.

The approach offers an inexpensive local search technique for these problems.

Abstract

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.