Unbiased and Biased Variance-Reduced Forward-Reflected-Backward Splitting Methods for Stochastic Composite Inclusions

This paper develops new variance-reduction techniques for the forward-reflected-backward splitting (FRBS) method to solve a class of possibly nonmonotone stochastic composite inclusions. Unlike unbiased estimators such as mini-batching, developing stochastic biased variants faces a fundamental technical challenge and has not been utilized before for inclusions and fixed-point problems. We fill this gap by designing a new framework that can handle both unbiased and biased estimators. Our main idea is to construct stochastic variance-reduced estimators for the forward-reflected direction and use them to perform iterate updates. First, we propose a class of unbiased variance-reduced estimators and show that increasing mini-batch SGD, loopless-SVRG, and SAGA estimators fall within this class. For these unbiased estimators, we establish a O(1/k) best-iterate convergence rate for the expected squared residual norm, together with almost-sure convergence of the iterate sequence to a solution. Consequently, we prove that the best oracle complexities for the n-finite-sum and expectation settings are O(n 2/3ε-2) and O(ε-10/3), respectively, when employing loopless-SVRG or SAGA, where ε is a desired accuracy. Second, we introduce a new class of biased variance-reduced estimators for the forward-reflected direction, which includes SARAH, Hybrid SGD, and Hybrid SVRG as special instances. While the convergence rates remain valid for these biased estimators, the resulting oracle complexities are O(n 3/4ε-2) and O(ε-5) for the n-finite-sum and expectation settings, respectively. Finally, we conduct two numerical experiments on AUC optimization for imbalanced classification and policy evaluation in reinforcement learning.