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

2026-03-16

AI Summary

This work addresses stochastic composite inclusion problems that may be non-monotone, particularly tackling the challenge of lacking effective variance reduction methods under biased estimators. The authors propose a unified framework that, for the first time, introduces biased variance-reduced estimators to inclusion and fixed-point problems, designing a new class of estimators tailored for the forward-reflected-backward splitting algorithm and providing a unified analysis covering both unbiased and biased settings. By integrating variance reduction techniques such as loopless-SVRG and SAGA, the method achieves an expected residual convergence rate of O(1/k) and almost sure convergence in the unbiased case, with oracle complexities of O(n^{2/3}ε^{-2}) and O(ε^{-10/3}), respectively. In the biased setting, the corresponding complexities are O(n^{3/4}ε^{-2}) and O(ε^{-5}). The approach is validated through applications in AUC optimization and policy evaluation in reinforcement learning.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

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.

Problem

• stochastic composite inclusions
• variance reduction
• forward-reflected-backward splitting
• biased estimators
• nonmonotone operators

Innovation

• variance reduction
• forward-reflected-backward splitting
• biased estimators
• stochastic composite inclusions
• oracle complexity