# Stochastic variance reduced extragradient methods for solving hierarchical variational inequalities

arXiv: https://arxiv.org/abs/2602.13510 · HTML: https://mlsanity.com/paper/2602.13510/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 58 references that are in the map.
- First posted on arXiv 13 Feb 2026.

## Builds on (most cited first, 27 of 27)

- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [On the convergence of single-call stochastic extra-gradient methods](https://mlsanity.com/paper/1908.08465/) (2019)
- [Reducing Noise in GAN Training with Variance Reduced Extragradient](https://mlsanity.com/paper/1904.08598/) (2019)
- [Error bounds, quadratic growth, and linear convergence of proximal methods](https://mlsanity.com/paper/1602.06661/) (2016)
- [Stochastic Variance Reduction Methods for Saddle-Point Problems](https://mlsanity.com/paper/1605.06398/) (2016)
- [Stochastic Variance Reduction for Variational Inequality Methods](https://mlsanity.com/paper/2102.08352/) (2021)
- [On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis](https://mlsanity.com/paper/2301.00712/) (2023)
- [Golden Ratio Algorithms for Variational Inequalities](https://mlsanity.com/paper/1803.08832/) (2018)
- [Simple and optimal methods for stochastic variational inequalities, I: operator extrapolation](https://mlsanity.com/paper/2011.02987/) (2020)
- [Extragradient method with variance reduction for stochastic variational inequalities](https://mlsanity.com/paper/1703.00260/) (2017)
- [Stability and Generalization of Bilevel Programming in Hyperparameter Optimization](https://mlsanity.com/paper/2106.04188/) (2021)
- [Stochastic Extragradient: General Analysis and Improved Rates](https://mlsanity.com/paper/2111.08611/) (2021)
- [Projection-Free Methods for Stochastic Simple Bilevel Optimization with Convex Lower-level Problem](https://mlsanity.com/paper/2308.07536/) (2023)
- [The Last-Iterate Convergence Rate of Optimistic Mirror Descent in Stochastic Variational Inequalities](https://mlsanity.com/paper/2107.01906/) (2021)
- [Smooth Monotone Stochastic Variational Inequalities and Saddle Point Problems: A Survey](https://mlsanity.com/paper/2208.13592/) (2022)
- [Penalty-based Methods for Simple Bilevel Optimization under H"olderian Error Bounds](https://mlsanity.com/paper/2402.02155/) (2024)
- [First-Order Methods for Convex Optimization](https://mlsanity.com/paper/2101.00935/) (2021)
- [Stochastic Approximation for Estimating the Price of Stability in Stochastic Nash Games](https://mlsanity.com/paper/2203.01271/) (2022)
- [Convergence of Extragradient SVRG for Variational Inequalities: Error Bounds and Increasing Iterate Averaging](https://mlsanity.com/paper/2306.01796/) (2023)
- [Improved guarantees for optimal Nash equilibrium seeking and bilevel variational inequalities](https://mlsanity.com/paper/2307.12511/) (2023)
- [Regularized Operator Extrapolation Method For Stochastic Bilevel Variational Inequality Problems](https://mlsanity.com/paper/2505.09778/) (2025)
- [Forward-Backward and Tseng's Type Penalty Schemes for Monotone Inclusion Problems](https://mlsanity.com/paper/1306.0352/) (2013)
- [On the solution of monotone nested variational inequalities](https://mlsanity.com/paper/2105.13463/) (2021)
- [Hierarchical Variational Inequality Problem for Noncooperative Game-Theoretic Selection of Generalized Nash Equilibrium](https://mlsanity.com/paper/2504.03208/) (2025)
- [Accelerating Diagonal Methods for Bilevel Optimization: Unified Convergence via Continuous-Time Dynamics](https://mlsanity.com/paper/2505.14389/) (2025)
- [Extragradient methods with complexity guarantees for hierarchical variational inequalities](https://mlsanity.com/paper/2512.20791/) (2025)

## Built on by (most cited first, 1 of 1)

- [Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems](https://mlsanity.com/paper/2605.08006/) (2026)

## Abstract (arXiv)

We are concerned with optimization in a broad sense through the lens of solving variational inequalities (VIs) -- a class of problems that are so general that they cover as particular cases minimization of functions, saddle-point (minimax) problems, Nash equilibrium problems, and many others. The key challenges in our problem formulation are the two-level hierarchical structure and finite-sum representation of the smooth operators in each level. For this setting, we are the first to prove convergence rates and complexity statements for variance-reduced stochastic algorithms approaching the solution of hierarchical VIs in Euclidean and Bregman setups.