# Last-Iterate Guarantees for Learning in Co-coercive Games

arXiv: https://arxiv.org/abs/2604.19065 · HTML: https://mlsanity.com/paper/2604.19065/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 22 references that are in the map.
- First posted on arXiv 21 Apr 2026.

## Builds on (most cited first, 15 of 15)

- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [On Gradient-Based Learning in Continuous Games](https://mlsanity.com/paper/1804.05464/) (2018)
- [Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity](https://mlsanity.com/paper/2107.00052/) (2021)
- [Bandit learning in concave N-person games](https://mlsanity.com/paper/1810.01925/) (2018)
- [Last-Iterate Convergence of Optimistic Gradient Method for Monotone Variational Inequalities](https://mlsanity.com/paper/2205.08446/) (2022)
- [Finite-Time Last-Iterate Convergence for Multi-Agent Learning in Games](https://mlsanity.com/paper/2002.09806/) (2020)
- [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)
- [Convergence rate analysis of several splitting schemes](https://mlsanity.com/paper/1406.4834/) (2014)
- [Learning Nash Equilibria in Monotone Games](https://mlsanity.com/paper/1904.01882/) (2019)
- [Convergence Rates with Inexact Non-expansive Operators](https://mlsanity.com/paper/1404.4837/) (2014)
- [Geometric Convergence of Gradient Play Algorithms for Distributed Nash Equilibrium Seeking](https://mlsanity.com/paper/1809.07383/) (2018)
- [On the Rate of Convergence of Payoff-based Algorithms to Nash Equilibrium in Strongly Monotone Games](https://mlsanity.com/paper/2202.11147/) (2022)
- [Learning to Control Unknown Strongly Monotone Games](https://mlsanity.com/paper/2407.00575/) (2024)
- [Convergence Rate of Generalized Nash Equilibrium Learning in Strongly Monotone Games with Linear Constraints](https://mlsanity.com/paper/2507.12112/) (2025)

## Abstract (arXiv)

We establish finite-time last-iterate guarantees for vanilla stochastic gradient descent in co-coercive games under noisy feedback. This is a broad class of games that is more general than strongly monotone games, allows for multiple Nash equilibria, and includes examples such as quadratic games with negative semidefinite interaction matrices and potential games with smooth concave potentials. Prior work in this setting has relied on relative noise models, where the noise vanishes as iterates approach equilibrium, an assumption that is often unrealistic in practice. We work instead under a substantially more general noise model in which the second moment of the noise is allowed to scale affinely with the squared norm of the iterates, an assumption natural in learning with unbounded action spaces. Under this model, we prove a last-iterate bound of order O(log(t)/t1/3), the first such bound for co-coercive games under non-vanishing noise. We additionally establish almost sure convergence of the iterates to the set of Nash equilibria and derive time-average convergence guarantees.