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 $\mathcal{O}(1/k)$ 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.

A Class of Accelerated Fixed-Point-Based Methods with Delayed Inexact Oracles and Its Applications

In this paper, we develop a novel accelerated fixed-point-based framework using delayed inexact oracles to approximate a fixed point of a nonexpansive operator (or equivalently, a root of a co-coercive operator), a central problem in scientific computing. Our approach leverages both Nesterov's acceleration technique and the Krasnosel'skii-Mann (KM) iteration, while accounting for delayed inexact oracles, a key mechanism in asynchronous algorithms. We also introduce a unified approximate error condition for delayed inexact oracles, which can cover various practical scenarios. Under mild conditions and appropriate parameter updates, we establish both $\mathcal{O}(1/k^2)$ non-asymptotic and $o(1/k^2)$ asymptotic convergence rates in expectation for the squared norm of residual. Our rate significantly improves the $\mathcal{O}(1/k)$ rates in classical KM-type methods, including their asynchronous variants. We also establish $o(1/k^2)$ almost sure convergence rates and the almost sure convergence of iterates to a solution of the problem. Within our framework, we instantiate three settings for the underlying operator: (i) a deterministic universal delayed oracle; (ii) a stochastic delayed oracle; and (iii) a finite-sum structure with asynchronous updates. For each case, we instantiate our framework to obtain a concrete algorithmic variant for which our convergence results still apply, and whose iteration complexity depends linearly on the maximum delay. Finally, we verify our algorithms and theoretical results through two numerical examples on both matrix game and shallow neural network training problems.

Accelerated Extragradient-Type Methods -- Part 2: Generalization and Sublinear Convergence Rates under Co-Hypomonotonicity

Following the first part of our project, this paper comprehensively studies two types of extragradient-based methods: anchored extragradient and Nesterov's accelerated extragradient for solving [non]linear inclusions (and, in particular, equations), primarily under the Lipschitz continuity and the co-hypomonotonicity assumptions. We unify and generalize a class of anchored extragradient methods for monotone inclusions to a wider range of schemes encompassing existing algorithms as special cases. We establish $\mathcal{O}(1/k)$ last-iterate convergence rates on the residual norm of the underlying mapping for this general framework and then specialize it to obtain convergence guarantees for specific instances, where $k$ denotes the iteration counter. We extend our approach to a class of anchored Tseng's forward-backward-forward splitting methods to obtain a broader class of algorithms for solving co-hypomonotone inclusions. Again, we analyze $\mathcal{O}(1/k)$ last-iterate convergence rates for this general scheme and specialize it to obtain convergence results for existing and new variants. We generalize and unify Nesterov's accelerated extra-gradient method to a new class of algorithms that covers existing schemes as special instances while generating new variants. For these schemes, we can prove $\mathcal{O}(1/k)$ last-iterate convergence rates for the residual norm under co-hypomonotonicity, covering a class of nonmonotone problems. We propose another novel class of Nesterov's accelerated extragradient methods to solve inclusions. Interestingly, these algorithms achieve both $\mathcal{O}(1/k)$ and $o(1/k)$ last-iterate convergence rates, and also the convergence of iterate sequences under co-hypomonotonicity and Lipschitz continuity. Finally, we provide a set of numerical experiments encompassing different scenarios to validate our algorithms and theoretical guarantees.

Revisiting Extragradient-Type Methods -- Part 1: Generalizations and Sublinear Convergence Rates

This paper presents a comprehensive analysis of the well-known extragradient (EG) method for solving both equations and inclusions. First, we unify and generalize EG for [non]linear equations to a wider class of algorithms, encompassing various existing schemes and potentially new variants. Next, we analyze both sublinear ``best-iterate'' and ``last-iterate'' convergence rates for the entire class of algorithms, and derive new convergence results for two well-known instances. Second, we extend our EG framework above to ``monotone'' inclusions, introducing a new class of algorithms and its corresponding convergence results. Third, we also unify and generalize Tseng's forward-backward-forward splitting (FBFS) method to a broader class of algorithms to solve [non]linear inclusions when a weak-Minty solution exists, and establish its ``best-iterate'' convergence rate. Fourth, to complete our picture, we also investigate sublinear rates of two other common variants of EG using our EG analysis framework developed here: the reflected forward-backward splitting and the golden ratio methods. Finally, we conduct an extensive numerical experiment to validate our theoretical findings. Our results demonstrate that several new variants of our proposed algorithms outperform existing schemes in the majority of examples.