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A Unifying Framework for Variance-Reduced Algorithms for Findings Zeroes of Monotone operators

Xun Zhang, William B. Haskell, Zhisheng Ye; 23(60):1−44, 2022.

Abstract

It is common to encounter large-scale monotone inclusion problems where the objective has a finite sum structure. We develop a general framework for variance-reduced forward-backward splitting algorithms for this problem. This framework includes a number of existing deterministic and variance-reduced algorithms for function minimization as special cases, and it is also applicable to more general problems such as saddle-point problems and variational inequalities. With a carefully constructed Lyapunov function, we show that the algorithms covered by our framework enjoy a linear convergence rate in expectation under mild assumptions. We further consider Catalyst acceleration and asynchronous implementation to reduce the algorithmic complexity and computation time. We apply our proposed framework to a policy evaluation problem and a strongly monotone two-player game, both of which fall outside the realm of function minimization.

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