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

• URL: http://arxiv.org/abs/2603.15576v1
• Date: Mon, 16 Mar 2026 17:39:25 GMT
• Title: Unbiased and Biased Variance-Reduced Forward-Reflected-Backward Splitting Methods for Stochastic Composite Inclusions
• Abstract summary: We develop new variance-reduction techniques for the forward-reflected-backward splitting (FRBS) method.<n>Unlike unbiased estimators such as mini-batching, developing biased variants faces a fundamental technical challenge.<n>We show that the best oracle complexities for the $n$-finite-sum and expectation settings are $mathcalO(n2/3-2)$ and $mathcalO(-10/3)$, respectively, when employing loopless-SVRG or SAGA.
• Score: 3.6997773420183866
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\mathcal{O}(1/k)$ 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 $\mathcal{O}(n^{2/3}ε^{-2})$ and $\mathcal{O}(ε^{-10/3})$, 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 $\mathcal{O}(n^{3/4}ε^{-2})$ and $\mathcal{O}(ε^{-5})$ 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.

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