# Simple and optimal methods for stochastic variational inequalities, I: operator extrapolation

arXiv: https://arxiv.org/abs/2011.02987 · HTML: https://mlsanity.com/paper/2011.02987/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2020.

## Built on by (most cited first, 30 of 35)

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- [Perseus: A Simple and Optimal High-Order Method for Variational Inequalities](https://mlsanity.com/paper/2205.03202/) (2022)
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- [A Fully Parameter-Free Second-Order Algorithm for Convex-Concave Minimax Problems](https://mlsanity.com/paper/2407.03571/) (2024)
- [Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation](https://mlsanity.com/paper/2306.16617/) (2023)
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- [Multiscale replay: A robust algorithm for stochastic variational inequalities with a Markovian buffer](https://mlsanity.com/paper/2601.01502/) (2026)
- [Finding a stationary point of a stochastic convex problem](https://mlsanity.com/paper/2607.06883/) (2026)
- [How to Make the Gradient Mapping Small for Constrained Stochastic Min-Max Problems and Beyond](https://mlsanity.com/paper/2609.08380/) (2026)
- [An alternative approach to train neural networks using monotone variational inequality](https://mlsanity.com/paper/2202.08876/) (2022)
- [Primal-dual extrapolation methods for monotone inclusions under local Lipschitz continuity](https://mlsanity.com/paper/2206.00973/) (2022)
- [Reinforcement Learning for SBM Graphon Games with Re-Sampling](https://mlsanity.com/paper/2310.16326/) (2023)
- [Accelerated Distributional Temporal Difference Learning with Linear Function Approximation](https://mlsanity.com/paper/2511.12688/) (2025)
- [Stochastic variance reduced extragradient methods for solving hierarchical variational inequalities](https://mlsanity.com/paper/2602.13510/) (2026)
- [Value Mirror Descent for Reinforcement Learning](https://mlsanity.com/paper/2604.06039/) (2026)

## Abstract (arXiv)

In this paper we first present a novel operator extrapolation (OE) method for solving deterministic variational inequality (VI) problems. Similar to the gradient (operator) projection method, OE updates one single search sequence by solving a single projection subproblem in each iteration. We show that OE can achieve the optimal rate of convergence for solving a variety of VI problems in a much simpler way than existing approaches. We then introduce the stochastic operator extrapolation (SOE) method and establish its optimal convergence behavior for solving different stochastic VI problems. In particular, SOE achieves the optimal complexity for solving a fundamental problem, i.e., stochastic smooth and strongly monotone VI, for the first time in the literature. We also present a stochastic block operator extrapolations (SBOE) method to further reduce the iteration cost for the OE method applied to large-scale deterministic VIs with a certain block structure. Numerical experiments have been conducted to demonstrate the potential advantages of the proposed algorithms. In fact, all these algorithms are applied to solve generalized monotone variational inequality (GMVI) problems whose operator is not necessarily monotone. We will also discuss optimal OE-based policy evaluation methods for reinforcement learning in a companion paper.