# Accelerated and Stable Convergence with Anchored Generalized Optimistic Method

arXiv: https://arxiv.org/abs/2606.21528 · HTML: https://mlsanity.com/paper/2606.21528/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 35 of its 51 references that are in the map.
- First posted on arXiv 19 Jun 2026.

## Builds on (most cited first, 30 of 35)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Responsive Safety in Reinforcement Learning by PID Lagrangian Methods](https://mlsanity.com/paper/2007.03964/) (2020)
- [A Variational Inequality Perspective on Generative Adversarial Networks](https://mlsanity.com/paper/1802.10551/) (2018)
- [A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach](https://mlsanity.com/paper/1901.08511/) (2019)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [On the convergence of single-call stochastic extra-gradient methods](https://mlsanity.com/paper/1908.08465/) (2019)
- [Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with O(1/k2) Rate on Squared Gradient Norm](https://mlsanity.com/paper/2102.07922/) (2021)
- [A Unified Approach to Reinforcement Learning, Quantal Response Equilibria, and Two-Player Zero-Sum Games](https://mlsanity.com/paper/2206.05825/) (2022)
- [Fast Extra Gradient Methods for Smooth Structured Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2106.02326/) (2021)
- [High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance](https://mlsanity.com/paper/2302.00999/) (2023)
- [Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities](https://mlsanity.com/paper/2002.08872/) (2020)
- [Negative Momentum for Improved Game Dynamics](https://mlsanity.com/paper/1807.04740/) (2018)
- [Explore Aggressively, Update Conservatively: Stochastic Extragradient Methods with Variable Stepsize Scaling](https://mlsanity.com/paper/2003.10162/) (2020)
- [Tight last-iterate convergence rates for no-regret learning in multi-player games](https://mlsanity.com/paper/2010.13724/) (2020)
- [On the Ineffectiveness of Variance Reduced Optimization for Deep Learning](https://mlsanity.com/paper/1812.04529/) (2018)
- [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)
- [Last-Iterate Convergence of Optimistic Gradient Method for Monotone Variational Inequalities](https://mlsanity.com/paper/2205.08446/) (2022)
- [Accelerated Single-Call Methods for Constrained Min-Max Optimization](https://mlsanity.com/paper/2210.03096/) (2022)
- [Finite Regret and Cycles with Fixed Step-Size via Alternating Gradient Descent-Ascent](https://mlsanity.com/paper/1907.04392/) (2019)
- [High-Probability Convergence for Composite and Distributed Stochastic Minimization and Variational Inequalities with Heavy-Tailed Noise](https://mlsanity.com/paper/2310.01860/) (2023)
- [Halpern-Type Accelerated and Splitting Algorithms For Monotone Inclusions](https://mlsanity.com/paper/2110.08150/) (2021)
- [Near-Optimal Algorithms for Making the Gradient Small in Stochastic Minimax Optimization](https://mlsanity.com/paper/2208.05925/) (2022)
- [Stochastic Halpern Iteration with Variance Reduction for Stochastic Monotone Inclusions](https://mlsanity.com/paper/2203.09436/) (2022)
- [Tight Last-Iterate Convergence of the Extragradient and the Optimistic Gradient Descent-Ascent Algorithm for Constrained Monotone Variational Inequalities](https://mlsanity.com/paper/2204.09228/) (2022)
- [On PI Controllers for Updating Lagrange Multipliers in Constrained Optimization](https://mlsanity.com/paper/2406.04558/) (2024)
- [Accelerating Value Iteration with Anchoring](https://mlsanity.com/paper/2305.16569/) (2023)
- [Smooth Monotone Stochastic Variational Inequalities and Saddle Point Problems: A Survey](https://mlsanity.com/paper/2208.13592/) (2022)
- [Optimal Acceleration for Minimax and Fixed-Point Problems is Not Unique](https://mlsanity.com/paper/2404.13228/) (2024)

## Abstract (arXiv)

We study first-order methods for solving monotone variational inequalities arising in min-max optimization. Classical approaches such as the extragradient method rely on two gradient queries per iteration, which limits their analysis and applicability in the online and stochastic settings. We propose a family of Generalized Optimistic Methods with Anchoring (GOMA), which combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration. In the deterministic setting, GOMA achieves the optimal accelerated last-iterate rate O(1/k2) on the squared gradient norm for monotone Lipschitz operators. In the stochastic setting with unbounded variance, a simplified single-call variant of GOMA achieves a last-iterate convergence rate of O(1/√k) on the squared gradient norm. To the best of our knowledge, this is the first such guarantee for stochastic monotone Lipschitz variational inequalities in the unconstrained setting without variance reduction or growing batches.