# Controlled-Minimax: Stabilizing Adversarial Optimization via Control Theory

arXiv: https://arxiv.org/abs/2610.12915 · HTML: https://mlsanity.com/paper/2610.12915/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 35 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 19 of 19)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes](https://mlsanity.com/paper/1805.08114/) (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)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [The Numerics of GANs](https://mlsanity.com/paper/1705.10461/) (2017)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Gradient descent GAN optimization is locally stable](https://mlsanity.com/paper/1706.04156/) (2017)
- [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)
- [Negative Momentum for Improved Game Dynamics](https://mlsanity.com/paper/1807.04740/) (2018)
- [Finding Mixed Nash Equilibria of Generative Adversarial Networks](https://mlsanity.com/paper/1811.02002/) (2018)
- [Stabilizing GANs' Training with Brownian Motion Controller](https://mlsanity.com/paper/2306.10468/) (2023)

## Abstract (arXiv)

Minimax optimization is fundamental to adversarial learning yet suffers from severe stability challenges, often manifesting as persistent limit cycles or divergence. This paper investigates these instabilities through the lens of continuous-time dynamical systems. By modeling training dynamics as a second-order Ordinary Differential Equation (ODE), we rigorously identify the root cause of non-convergence: the problematic eigenstructure of the loss function's Hessian near equilibrium points. To address this, we introduce Controlled-Minimax, a novel stabilization framework that reformulates the optimization trajectory as a controllable physical system. Unlike heuristic regularization techniques, our approach derives a specific feedback controller that provides theoretical guarantees of local exponential stability, regardless of the underlying Hessian spectrum. Consequently, our method stabilizes general minimax problems where traditional gradient descent-ascent methods fail. Extensive experiments on pathological benchmarks including the Dirac-GAN, 2D Gaussian mixtures, and Swiss Roll datasets, as well as high-resolution image synthesis with StyleGAN2-ada, demonstrate that Controlled-Minimax effectively suppresses oscillatory behavior while ensuring fast and robust convergence.