We consider monotone inclusion problems where the operators may be expectation-valued, a class of problems that subsumes convex stochastic optimization problems 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 that is addressed by using sampling. Accordingly, we propose an avenue for addressing uncertainty in the mapping: Variance-reduced stochastic modified forward-backward splitting scheme (vr-SMFBS). In constrained settings, 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 O(1/k) (monotone A) rates of convergence while achieving optimal oracle complexity bounds. The rate statements in monotone regimes appear to be amongst the first and rely on leveraging 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.
翻译:我们考虑单调包容问题,因为操作者可能会得到预期价值的估价,这是一组问题,其次变相优化问题,以及随机变化性不平等和均衡问题的细类。直接应用分裂计划由于需要解决每步都使用预期价值地图的问题而变得复杂,这是通过抽样处理的一个问题。因此,我们建议了一种解决绘图不确定性的途径:差异性差异减少的随机调整后后向后向分解计划(vr-SMFBS)。在受限制的情况下,我们考虑结构化的设置,当地图可以分解成期待值地图A和最大单调图B,并具有可移动的固态。我们表明,拟议的计划配备了a.s. 趋同保证,线性(强单调A)和O(monotone A) 趋同率,同时达到最佳或最接近的复杂程度。单调制制度的利率似乎属于第一种类别,并且依赖利用菲特差差差功能实现单调的地图A和最大单调 B地图B图B。我们表明,在稳定性规则中,稳定度要求更弱的平极化的平极化方法可以反映。