# Nesterov Meets Optimism: Rate-Optimal Separable Minimax Optimization

arXiv: https://arxiv.org/abs/2210.17550 · HTML: https://mlsanity.com/paper/2210.17550/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 57 references that are in the map.
- First posted on arXiv 31 Oct 2022.

## Builds on (most cited first, 30 of 32)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [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)
- [Near-Optimal Algorithms for Minimax Optimization](https://mlsanity.com/paper/2002.02417/) (2020)
- [Provable Self-Play Algorithms for Competitive Reinforcement Learning](https://mlsanity.com/paper/2002.04017/) (2020)
- [On the convergence of single-call stochastic extra-gradient methods](https://mlsanity.com/paper/1908.08465/) (2019)
- [Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems](https://mlsanity.com/paper/1808.02901/) (2018)
- [Interaction Matters: A Note on Non-asymptotic Local Convergence of Generative Adversarial Networks](https://mlsanity.com/paper/1802.06132/) (2018)
- [Reducing Noise in GAN Training with Variance Reduced Extragradient](https://mlsanity.com/paper/1904.08598/) (2019)
- [Stochastic Variance Reduction Methods for Saddle-Point Problems](https://mlsanity.com/paper/1605.06398/) (2016)
- [Stochastic Variance Reduction Methods for Policy Evaluation](https://mlsanity.com/paper/1702.07944/) (2017)
- [Tight last-iterate convergence rates for no-regret learning in multi-player games](https://mlsanity.com/paper/2010.13724/) (2020)
- [Stochastic Variance Reduction for Variational Inequality Methods](https://mlsanity.com/paper/2102.08352/) (2021)
- [The Complexity of Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2103.15888/) (2021)
- [Improved Algorithms for Convex-Concave Minimax Optimization](https://mlsanity.com/paper/2006.06359/) (2020)
- [Last-Iterate Convergence of Optimistic Gradient Method for Monotone Variational Inequalities](https://mlsanity.com/paper/2205.08446/) (2022)
- [Accelerated Schemes For A Class of Variational Inequalities](https://mlsanity.com/paper/1403.4164/) (2014)
- [Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity](https://mlsanity.com/paper/1802.01504/) (2018)
- [On Lower Iteration Complexity Bounds for the Saddle Point Problems](https://mlsanity.com/paper/1912.07481/) (2019)
- [Accelerating Smooth Games by Manipulating Spectral Shapes](https://mlsanity.com/paper/2001.00602/) (2020)
- [Relative Lipschitzness in Extragradient Methods and a Direct Recipe for Acceleration](https://mlsanity.com/paper/2011.06572/) (2020)
- [On the Convergence of Stochastic Extragradient for Bilinear Games using Restarted Iteration Averaging](https://mlsanity.com/paper/2107.00464/) (2021)
- [Accelerated Primal-Dual Gradient Method for Smooth and Convex-Concave Saddle-Point Problems with Bilinear Coupling](https://mlsanity.com/paper/2112.15199/) (2021)
- [Sharper Rates for Separable Minimax and Finite Sum Optimization via Primal-Dual Extragradient Methods](https://mlsanity.com/paper/2202.04640/) (2022)
- [A Simple Nearly-Optimal Restart Scheme For Speeding-Up First Order Methods](https://mlsanity.com/paper/1803.00151/) (2018)
- [Lifted Primal-Dual Method for Bilinearly Coupled Smooth Minimax Optimization](https://mlsanity.com/paper/2201.07427/) (2022)
- [Robust Accelerated Primal-Dual Methods for Computing Saddle Points](https://mlsanity.com/paper/2111.12743/) (2021)
- [Smooth Monotone Stochastic Variational Inequalities and Saddle Point Problems: A Survey](https://mlsanity.com/paper/2208.13592/) (2022)
- [DIPPA: An improved Method for Bilinear Saddle Point Problems](https://mlsanity.com/paper/2103.08270/) (2021)

## Built on by (most cited first, 2 of 2)

- [Accelerated Preference Optimization for Large Language Model Alignment](https://mlsanity.com/paper/2410.06293/) (2024)
- [Generalized Continuous-Time Models for Nesterov's Accelerated Gradient Methods](https://mlsanity.com/paper/2409.00913/) (2024)

## Abstract (arXiv)

We propose a new first-order optimization algorithm -- AcceleratedGradient-OptimisticGradient (AG-OG) Descent Ascent -- for separable convex-concave minimax optimization. The main idea of our algorithm is to carefully leverage the structure of the minimax problem, performing Nesterov acceleration on the individual component and optimistic gradient on the coupling component. Equipped with proper restarting, we show that AG-OG achieves the optimal convergence rate (up to a constant) for a variety of settings, including bilinearly coupled strongly convex-strongly concave minimax optimization (bi-SC-SC), bilinearly coupled convex-strongly concave minimax optimization (bi-C-SC), and bilinear games. We also extend our algorithm to the stochastic setting and achieve the optimal convergence rate in both bi-SC-SC and bi-C-SC settings. AG-OG is the first single-call algorithm with optimal convergence rates in both deterministic and stochastic settings for bilinearly coupled minimax optimization problems.