A Unifying Framework for Variance Reduction Algorithms for Finding Zeroes of Monotone Operators
This work provides a general framework for variance reduction in monotone operator problems, applicable to domains like saddle-point problems and variational inequalities, but it is incremental as it builds on existing algorithms.
The authors tackled large-scale monotone inclusion problems with finite sum structures by developing a unifying framework for variance-reduced forward-backward splitting algorithms, achieving a linear convergence rate in expectation under mild assumptions.
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 of function minimization.