Convergence analysis of the stochastic reflected forward–backward splitting algorithm

We propose and analyze the convergence of a novel stochastic algorithm for solving monotone inclusions that are the sum of a maximal monotone operator and a monotone, Lipschitzian operator. The propose algorithm requires only unbiased estimations of the Lipschitzian operator. We obtain the rate O (...

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Bibliographic Details
Published in:Optimization letters Vol. 16; no. 9; pp. 2649 - 2679
Main Authors: Nguyen, Van Dung, Vũ, Bắng Công
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2022
Subjects:
Computational Intelligence
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Operations Research/Decision Theory
Optimization
Original Paper
Simulation
Monotone inclusion
Duality
Stochastic optimization
Reflected method
Ergodic convergence
Primal–dual algorithm
ISSN:1862-4472, 1862-4480
Online Access:Get full text
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Summary:We propose and analyze the convergence of a novel stochastic algorithm for solving monotone inclusions that are the sum of a maximal monotone operator and a monotone, Lipschitzian operator. The propose algorithm requires only unbiased estimations of the Lipschitzian operator. We obtain the rate O ( l o g ( n ) / n ) in expectation for the strongly monotone case, as well as almost sure convergence for the general case. Furthermore, in the context of application to convex–concave saddle point problems, we derive the rate of the primal–dual gap. In particular, we also obtain O ( 1 / n ) rate convergence of the primal–dual gap in the deterministic setting.
ISSN:1862-4472
1862-4480
DOI:10.1007/s11590-021-01844-8