# Two-timescale Extragradient for Finding Local Minimax Points

arXiv: https://arxiv.org/abs/2305.16242 · HTML: https://mlsanity.com/paper/2305.16242/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 45 references that are in the map.
- First posted on arXiv 25 May 2023.

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

- [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)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [NIPS 2016 Tutorial: Generative Adversarial Networks](https://mlsanity.com/paper/1701.00160/) (2016)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization](https://mlsanity.com/paper/2011.00364/) (2020)
- [On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach](https://mlsanity.com/paper/1910.07512/) (2019)
- [ODE Analysis of Stochastic Gradient Methods with Optimism and Anchoring for Minimax Problems](https://mlsanity.com/paper/1905.10899/) (2019)
- [Convergence of Proximal Point and Extragradient-Based Methods Beyond Monotonicity: the Case of Negative Comonotonicity](https://mlsanity.com/paper/2210.13831/) (2022)
- [Newton-type Methods for Minimax Optimization](https://mlsanity.com/paper/2006.14592/) (2020)
- [Tight Analysis of Extra-gradient and Optimistic Gradient Methods For Nonconvex Minimax Problems](https://mlsanity.com/paper/2210.09382/) (2022)
- [A Cubic Regularization Approach for Finding Local Minimax Points in Nonconvex Minimax Optimization](https://mlsanity.com/paper/2110.07098/) (2021)
- [Optimality and Stability in Non-Convex Smooth Games](https://mlsanity.com/paper/2002.11875/) (2020)

## Built on by (most cited first, 4 of 4)

- [Negative Stepsizes Make Gradient-Descent-Ascent Converge](https://mlsanity.com/paper/2505.01423/) (2025)
- [Properties of Fixed Points of Generalised Extra Gradient Methods Applied to Min-Max Problems](https://mlsanity.com/paper/2504.03069/) (2025)
- [Min-Max Optimization Is Strictly Easier Than Variational Inequalities](https://mlsanity.com/paper/2511.03052/) (2025)
- [On the Stability Connection Between Discrete-Time Algorithms and Their Resolution ODEs: Applications to Min-Max Optimisation](https://mlsanity.com/paper/2603.01430/) (2026)

## Abstract (arXiv)

Minimax problems are notoriously challenging to optimize. However, we present that the two-timescale extragradient method can be a viable solution. By utilizing dynamical systems theory, we show that it converges to points that satisfy the second-order necessary condition of local minimax points, under mild conditions that the two-timescale gradient descent ascent fails to work. This work provably improves upon all previous results on finding local minimax points, by eliminating a crucial assumption that the Hessian with respect to the maximization variable is nondegenerate.