# Last-Iterate Convergence of Optimistic Gradient Method for Monotone Variational Inequalities

arXiv: https://arxiv.org/abs/2205.08446 · HTML: https://mlsanity.com/paper/2205.08446/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2022.

## Built on by (most cited first, 30 of 33)

- [Accelerated Single-Call Methods for Constrained Min-Max Optimization](https://mlsanity.com/paper/2210.03096/) (2022)
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- [On the Convergence of No-Regret Learning Dynamics in Time-Varying Games](https://mlsanity.com/paper/2301.11241/) (2023)
- [Fast Last-Iterate Convergence of Learning in Games Requires Forgetful Algorithms](https://mlsanity.com/paper/2406.10631/) (2024)
- [Negative Stepsizes Make Gradient-Descent-Ascent Converge](https://mlsanity.com/paper/2505.01423/) (2025)
- [Instance-Dependent Regret Bounds for Learning Two-Player Zero-Sum Games with Bandit Feedback](https://mlsanity.com/paper/2502.17625/) (2025)
- [Adaptively Perturbed Mirror Descent for Learning in Games](https://mlsanity.com/paper/2305.16610/) (2023)
- [COMAL: A Convergent Meta-Algorithm for Aligning LLMs with General Preferences](https://mlsanity.com/paper/2410.23223/) (2024)
- [From Average-Iterate to Last-Iterate Convergence in Games: A Reduction and Its Applications](https://mlsanity.com/paper/2506.03464/) (2025)
- [Last-Iterate Convergence Properties of Regret-Matching Algorithms in Games](https://mlsanity.com/paper/2311.00676/) (2023)
- [Multiplayer Federated Learning: Reaching Equilibrium with Less Communication](https://mlsanity.com/paper/2501.08263/) (2025)
- [No-regret learning in harmonic games: Extrapolation in the face of conflicting interests](https://mlsanity.com/paper/2412.20203/) (2024)
- [Primal Methods for Variational Inequality Problems with Functional Constraints](https://mlsanity.com/paper/2403.12859/) (2024)
- [Efficient Inverse Multiagent Learning](https://mlsanity.com/paper/2502.14160/) (2025)
- [Unbiased and Biased Variance-Reduced Forward-Reflected-Backward Splitting Methods for Stochastic Composite Inclusions](https://mlsanity.com/paper/2603.15576/) (2026)
- [An Improved Last-Iterate Convergence Rate for Anchored Gradient Descent Ascent](https://mlsanity.com/paper/2604.03782/) (2026)
- [Nesterov Meets Optimism: Rate-Optimal Separable Minimax Optimization](https://mlsanity.com/paper/2210.17550/) (2022)
- [Optimistic Policy Gradient in Multi-Player Markov Games with a Single Controller: Convergence Beyond the Minty Property](https://mlsanity.com/paper/2312.12067/) (2023)
- [Diffusion Stochastic Optimization for Min-Max Problems](https://mlsanity.com/paper/2401.14585/) (2024)
- [On Separation Between Best-Iterate, Random-Iterate, and Last-Iterate Convergence of Learning in Games](https://mlsanity.com/paper/2503.02825/) (2025)
- [Distribution-informed Online Conformal Prediction](https://mlsanity.com/paper/2512.07770/) (2025)
- [Extragradient Type Methods for Riemannian Variational Inequality Problems](https://mlsanity.com/paper/2309.14155/) (2023)
- [Dual Optimistic Ascent (PI Control) is the Augmented Lagrangian Method in Disguise](https://mlsanity.com/paper/2509.22500/) (2025)
- [Shuffling the Data, Stretching the Step-size: Sharper Bias in constant step-size SGD](https://mlsanity.com/paper/2604.10373/) (2026)
- [Extragradient-Type Methods with O (1/k) Last-Iterate Convergence Rates for Co-Hypomonotone Inclusions](https://mlsanity.com/paper/2302.04099/) (2023)
- [Omega: Optimistic EMA Gradients](https://mlsanity.com/paper/2306.07905/) (2023)
- [Negative Momentum for Convex-Concave Optimization](https://mlsanity.com/paper/2604.17145/) (2026)
- [Last-Iterate Guarantees for Learning in Co-coercive Games](https://mlsanity.com/paper/2604.19065/) (2026)
- [A Unifying View of Anchoring via Operator-Side Tikhonov Regularization](https://mlsanity.com/paper/2605.30905/) (2026)
- [Accelerated and Stable Convergence with Anchored Generalized Optimistic Method](https://mlsanity.com/paper/2606.21528/) (2026)

## Abstract (arXiv)

The Past Extragradient (PEG) [Popov, 1980] method, also known as the Optimistic Gradient method, has known a recent gain in interest in the optimization community with the emergence of variational inequality formulations for machine learning. Recently, in the unconstrained case, Golowich et al. [2020] proved that a O(1/N) last-iterate convergence rate in terms of the squared norm of the operator can be achieved for Lipschitz and monotone operators with a Lipschitz Jacobian. In this work, by introducing a novel analysis through potential functions, we show that (i) this O(1/N) last-iterate convergence can be achieved without any assumption on the Jacobian of the operator, and (ii) it can be extended to the constrained case, which was not derived before even under Lipschitzness of the Jacobian. The proof is significantly different from the one known from Golowich et al. [2020], and its discovery was computer-aided. Those results close the open question of the last iterate convergence of PEG for monotone variational inequalities.