Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions

Patrick R. Johnstone, Pierre Moulin

Research output: Contribution to journalArticlepeer-review

Abstract

This paper is concerned with convex composite minimization problems in a Hilbert space. In these problems, the objective is the sum of two closed, proper, and convex functions where one is smooth and the other admits a computationally inexpensive proximal operator. We analyze a family of generalized inertial proximal splitting algorithms (GIPSA) for solving such problems. We establish weak convergence of the generated sequence when the minimum is attained. Our analysis unifies and extends several previous results. We then focus on ℓ1-regularized optimization, which is the ubiquitous special case where the nonsmooth term is the ℓ1-norm. For certain parameter choices, GIPSA is amenable to a local analysis for this problem. For these choices we show that GIPSA achieves finite “active manifold identification”, i.e. convergence in a finite number of iterations to the optimal support and sign, after which GIPSA reduces to minimizing a local smooth function. We prove local linear convergence under either restricted strong convexity or a strict complementarity condition. We determine the rate in terms of the inertia, stepsize, and local curvature. Our local analysis is applicable to certain recent variants of the Fast Iterative Shrinkage–Thresholding Algorithm (FISTA), for which we establish active manifold identification and local linear convergence. Based on our analysis we propose a momentum restart scheme in these FISTA variants to obtain the optimal local linear convergence rate while maintaining desirable global properties.

Original languageEnglish (US)
Pages (from-to)259-292
Number of pages34
JournalComputational Optimization and Applications
Volume67
Issue number2
DOIs
StatePublished - Jun 1 2017

Keywords

  • FISTA
  • Inertial forward-backward splitting
  • Inertial proximal gradient
  • Lasso
  • Local linear convergence
  • Momentum methods

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

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