Variance-Reduced Splitting Schemes for Monotone Stochastic Generalized Equations

We consider monotone inclusion problems with expectation-valued operators, a class of problems that subsumes convex stochastic optimization problems with possibly smooth expectation-valued constraints as well as subclasses of stochastic variational inequality and equilibrium problems. A direct application of splitting schemes is complicated by the need to resolve problems with expectation-valued maps at each step, a concern addressed via sampling. Accordingly, we propose an avenue for addressing uncertainty in the mapping: variance-reduced stochastic modified forward-backward splitting scheme. We consider structured settings when the map can be decomposed into an expectation-valued map $A$ and a maximal monotone map $B$ with a tractable resolvent. We show that the proposed schemes are equipped with a.s. convergence guarantees, linear (strongly monotone $A$) and $\mathcal {O}(1/k)$ (monotone $A$) rates of convergence while achieving optimal oracle complexity bounds. The rate statements in monotone regimes appear to be among the first and leverage the Fitzpatrick gap function for monotone inclusions. Furthermore, the schemes rely on weaker moment requirements on noise and allow for weakening unbiasedness requirements on oracles in strongly monotone regimes. Preliminary numerics on a class of two-stage stochastic variational inequality problems reflect these findings and show that the variance-reduced schemes outperform stochastic approximation schemes and sample-average approximation approaches. The benefits of attaining deterministic rates of convergence become even more salient when resolvent computation is expensive.