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On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence

Stephen Becker(stephen.becker***at***colorado.edu)
Jalal Fadili(Jalal.Fadili***at***ensicaen.fr)
Peter Ochs(ochs***at***math.uni-sb.de)

Abstract: We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal +/- rank-r symmetric positive definite matrices. This special type of metric allows for a highly efficient evaluation of the proximal mapping. The key to this efficiency is a general proximal calculus in the new metric. By using duality, formulas are derived that relate the proximal mapping in a rank-r modified metric to the original metric. We also describe efficient implementations of the proximity calculation for a large class of functions; the implementations exploit the piece-wise linear nature of the dual problem. Then, we apply these results to acceleration of composite convex minimization problems, which leads to elegant quasi-Newton methods for which we prove convergence. The algorithm is tested on several numerical examples and compared to a comprehensive list of alternatives in the literature. Our quasi-Newton splitting algorithm with the prescribed metric compares favorably against state-of-the-art. The algorithm has extensive applications including signal processing, sparse recovery, machine learning and classification to name a few.

Keywords: forward-backward splitting, quasi-Newton, proximal calculus, duality

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: arXiv:1801.08691

Download: [PDF]

Entry Submitted: 01/29/2018
Entry Accepted: 01/29/2018
Entry Last Modified: 01/29/2018

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