  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.
