Parallel random block-coordinate forward–backward algorithm: a unified convergence analysis

We study the block-coordinate forward–backward algorithm in which the blocks are updated in a random and possibly parallel manner, according to arbitrary probabilities. The algorithm allows different stepsizes along the block-coordinates to fully exploit the smoothness properties of the objective fu...

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Bibliographic Details
Published in:Mathematical programming Vol. 193; no. 1; pp. 225 - 269
Main Authors: Salzo, Saverio, Villa, Silvia
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2022
Springer
Springer Nature B.V
Subjects:
Algorithms
Analysis
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Convergence
Convexity
Error analysis
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Smoothness
Theoretical
Random block-coordinate descent
65K05
Convex optimization
90C25
Convergence rates
Error bounds
90C06
49M27
Arbitrary sampling
Stochastic quasi-Fejér sequences
Parallel algorithms
Forward–backward algorithm
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We study the block-coordinate forward–backward algorithm in which the blocks are updated in a random and possibly parallel manner, according to arbitrary probabilities. The algorithm allows different stepsizes along the block-coordinates to fully exploit the smoothness properties of the objective function. In the convex case and in an infinite dimensional setting, we establish almost sure weak convergence of the iterates and the asymptotic rate o (1/ n ) for the mean of the function values. We derive linear rates under strong convexity and error bound conditions. Our analysis is based on an abstract convergence principle for stochastic descent algorithms which allows to extend and simplify existing results.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-020-01602-1