# Newton-type Methods for Minimax Optimization

arXiv: https://arxiv.org/abs/2006.14592 · HTML: https://mlsanity.com/paper/2006.14592/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2020.

## Built on by (most cited first, 11 of 11)

- [Certified Minimax Unlearning with Generalization Rates and Deletion Capacity](https://mlsanity.com/paper/2312.10336/) (2023)
- [Indiscriminate Data Poisoning Attacks on Neural Networks](https://mlsanity.com/paper/2204.09092/) (2022)
- [Lifted Primal-Dual Method for Bilinearly Coupled Smooth Minimax Optimization](https://mlsanity.com/paper/2201.07427/) (2022)
- [Finding Second-Order Stationary Points in Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2110.04814/) (2021)
- [Complex Momentum for Optimization in Games](https://mlsanity.com/paper/2102.08431/) (2021)
- [Minimax Optimization with Smooth Algorithmic Adversaries](https://mlsanity.com/paper/2106.01488/) (2021)
- [A Cubic Regularization Approach for Finding Local Minimax Points in Nonconvex Minimax Optimization](https://mlsanity.com/paper/2110.07098/) (2021)
- [Two-timescale Extragradient for Finding Local Minimax Points](https://mlsanity.com/paper/2305.16242/) (2023)
- [HessianFR: An Efficient Hessian-based Follow-the-Ridge Algorithm for Minimax Optimization](https://mlsanity.com/paper/2205.11030/) (2022)
- [On the Stability Connection Between Discrete-Time Algorithms and Their Resolution ODEs: Applications to Min-Max Optimisation](https://mlsanity.com/paper/2603.01430/) (2026)
- [Min-Max Optimization Requires Exponentially Many Queries](https://mlsanity.com/paper/2605.13806/) (2026)

## Abstract (arXiv)

Differential games, in particular two-player sequential zero-sum games (a.k.a. minimax optimization), have been an important modeling tool in applied science and received renewed interest in machine learning due to many recent applications, such as adversarial training, generative models and reinforcement learning. However, existing theory mostly focuses on convex-concave functions with few exceptions. In this work, we propose two novel Newton-type algorithms for nonconvex-nonconcave minimax optimization. We prove their local convergence at strict local minimax points, which are surrogates of global solutions. We argue that our Newton-type algorithms nicely complement existing ones in that (a) they converge faster to strict local minimax points; (b) they are much more effective when the problem is ill-conditioned; (c) their computational complexity remains similar. We verify the effectiveness of our Newton-type algorithms through experiments on training GANs which are intrinsically nonconvex and ill-conditioned. Our code is available at https://github.com/watml/min-max-2nd-order.