Variance-Reduced Forward-Reflected Algorithms for Generalized Equations

Quoc Tran-Dinh

Keywords: Variance Reduction Method, SGD, Generalized Equation, Variational Inequality, Minimax Problem

TL;DR: This paper proposes novel variance-reduced methods for generalized equations with state-of-the-art oracle complexity

Abstract:

We develop two novel stochastic variance-reduction methods to approximate a solution of generalized equations applicable to both equations and inclusions. Our algorithms leverage a new combination of ideas from the forward-reflected-backward splitting method and a class of unbiased variance-reduced estimators. We construct two new stochastic estimators within this class, inspired by the well-known SVRG and SAGA estimators. These estimators significantly differ from existing approaches used in minimax and variational inequality problems. By appropriately selecting parameters, both algorithms achieve the state-of-the-art oracle complexity of $\mathcal{O}(n + n^{2/3} \epsilon^{-2})$ for obtaining an $\epsilon$-solution in terms of the operator residual norm, where $n$ represents the number of summands and $\epsilon$ signifies the desired accuracy. This complexity aligns with the best-known results in SVRG and SAGA methods for stochastic nonconvex optimization. We test our algorithms on two numerical examples and compare them with existing methods. The results demonstrate promising improvements offered by the new methods compared to their competitors.

Primary Area: optimization

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Submission Number: 10582