# Unbiased and Biased Variance-Reduced Forward-Reflected-Backward Splitting Methods for Stochastic Composite Inclusions arXiv: https://arxiv.org/abs/2603.15576 · HTML: https://mlsanity.com/paper/2603.15576/ ## Key facts - Cited by 3 papers in the map (data through 14 Oct 2026). - Builds on 47 of its 85 references that are in the map. - First posted on arXiv 16 Mar 2026. ## Builds on (most cited first, 30 of 47) - [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017) - [Multi-Agent Reinforcement Learning: A Selective Overview of Theories and Algorithms](https://mlsanity.com/paper/1911.10635/) (2019) - [Minimax Regret Bounds for Reinforcement Learning](https://mlsanity.com/paper/1703.05449/) (2017) - [SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives](https://mlsanity.com/paper/1407.0202/) (2014) - [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019) - [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018) - [SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient](https://mlsanity.com/paper/1703.00102/) (2017) - [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017) - [Large-Scale Methods for Distributionally Robust Optimization](https://mlsanity.com/paper/2010.05893/) (2020) - [Stochastic Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1603.06160/) (2016) - [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018) - [PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization](https://mlsanity.com/paper/2008.10898/) (2020) - [Random Reshuffling: Simple Analysis with Vast Improvements](https://mlsanity.com/paper/2006.05988/) (2020) - [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018) - [Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization](https://mlsanity.com/paper/2011.00364/) (2020) - [Fairness-aware Agnostic Federated Learning](https://mlsanity.com/paper/2010.05057/) (2020) - [Don't Jump Through Hoops and Remove Those Loops: SVRG and Katyusha are Better Without the Outer Loop](https://mlsanity.com/paper/1901.08689/) (2019) - [Reducing Noise in GAN Training with Variance Reduced Extragradient](https://mlsanity.com/paper/1904.08598/) (2019) - [Stochastic Variance Reduction Methods for Saddle-Point Problems](https://mlsanity.com/paper/1605.06398/) (2016) - [Stochastic Variance Reduction Methods for Policy Evaluation](https://mlsanity.com/paper/1702.07944/) (2017) - [Stochastic Variance Reduction for Variational Inequality Methods](https://mlsanity.com/paper/2102.08352/) (2021) - [Extragradient Method: O(1/K) Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity](https://mlsanity.com/paper/2110.04261/) (2021) - [Golden Ratio Algorithms for Variational Inequalities](https://mlsanity.com/paper/1803.08832/) (2018) - [Variance Reduction for Matrix Games](https://mlsanity.com/paper/1907.02056/) (2019) - [Global Convergence and Variance-Reduced Optimization for a Class of Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2002.09621/) (2020) - [Stochastic Gradient Descent-Ascent: Unified Theory and New Efficient Methods](https://mlsanity.com/paper/2202.07262/) (2022) - [A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity](https://mlsanity.com/paper/1808.04162/) (2018) - [Revisiting Stochastic Extragradient](https://mlsanity.com/paper/1905.11373/) (2019) - [Extragradient method with variance reduction for stochastic variational inequalities](https://mlsanity.com/paper/1703.00260/) (2017) - [Last-Iterate Convergence of Optimistic Gradient Method for Monotone Variational Inequalities](https://mlsanity.com/paper/2205.08446/) (2022) ## Built on by (most cited first, 3 of 3) - [How to Make the Gradient Mapping Small for Constrained Stochastic Min-Max Problems and Beyond](https://mlsanity.com/paper/2609.08380/) (2026) - [Improving the Last-Iterate Guarantees of Anytime Algorithms for Stochastic Monotone Variational Inequalities](https://mlsanity.com/paper/2609.15257/) (2026) - [Optimal Oracle Complexity for Finite-Sum Monotone Inclusions](https://mlsanity.com/paper/2610.05038/) (2026) ## Abstract (arXiv) 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(n2/3ε-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(n3/4ε-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.