Subgradient Methods for Sharp Weakly Convex Functions

Damek Davis, Dmitriy Drusvyatskiy (), Kellie J. MacPhee and Courtney Paquette ()
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Damek Davis: Cornell University
Dmitriy Drusvyatskiy: University of Washington
Kellie J. MacPhee: University of Washington
Courtney Paquette: Lehigh University

Journal of Optimization Theory and Applications, 2018, vol. 179, issue 3, No 11, 962-982

Abstract: 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.

Keywords: Subgradient; Weakly convex; Sharp; Error bound; Linear rate; 65K05; 65K10 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-018-1372-8

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