# Simultaneous Transport Evolution for Minimax Equilibria on Measures

arXiv: https://arxiv.org/abs/2202.06460 · HTML: https://mlsanity.com/paper/2202.06460/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 54 references that are in the map.
- First posted on arXiv 14 Feb 2022.

## Builds on (most cited first, 21 of 21)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Deep Decentralized Multi-task Multi-Agent Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/1703.06182/) (2017)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [Near-Optimal Algorithms for Minimax Optimization](https://mlsanity.com/paper/2002.02417/) (2020)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization](https://mlsanity.com/paper/1807.04252/) (2018)
- [The Complexity of Constrained Min-Max Optimization](https://mlsanity.com/paper/2009.09623/) (2020)
- [Interaction Matters: A Note on Non-asymptotic Local Convergence of Generative Adversarial Networks](https://mlsanity.com/paper/1802.06132/) (2018)
- [Finding Mixed Nash Equilibria of Generative Adversarial Networks](https://mlsanity.com/paper/1811.02002/) (2018)
- [Train simultaneously, generalize better: Stability of gradient-based minimax learners](https://mlsanity.com/paper/2010.12561/) (2020)
- [On Finding Local Nash Equilibria (and Only Local Nash Equilibria) in Zero-Sum Games](https://mlsanity.com/paper/1901.00838/) (2019)
- [Exponential Family Estimation via Adversarial Dynamics Embedding](https://mlsanity.com/paper/1904.12083/) (2019)
- [A mean-field analysis of two-player zero-sum games](https://mlsanity.com/paper/2002.06277/) (2020)
- [What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?](https://mlsanity.com/paper/1902.00618/) (2019)
- [Dual Training of Energy-Based Models with Overparametrized Shallow Neural Networks](https://mlsanity.com/paper/2107.05134/) (2021)

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

- [Two-Scale Gradient Descent Ascent Dynamics Finds Mixed Nash Equilibria of Continuous Games: A Mean-Field Perspective](https://mlsanity.com/paper/2212.08791/) (2022)
- [Provably convergent quasistatic dynamics for mean-field two-player zero-sum games](https://mlsanity.com/paper/2202.10947/) (2022)

## Abstract (arXiv)

Min-max optimization problems arise in several key machine learning setups, including adversarial learning and generative modeling. In their general form, in absence of convexity/concavity assumptions, finding pure equilibria of the underlying two-player zero-sum game is computationally hard [Daskalakis et al., 2021]. In this work we focus instead in finding mixed equilibria, and consider the associated lifted problem in the space of probability measures. By adding entropic regularization, our main result establishes global convergence towards the global equilibrium by using simultaneous gradient ascent-descent with respect to the Wasserstein metric -- a dynamics that admits efficient particle discretization in high-dimensions, as opposed to entropic mirror descent. We complement this positive result with a related entropy-regularized loss which is not bilinear but still convex-concave in the Wasserstein geometry, and for which simultaneous dynamics do not converge yet timescale separation does. Taken together, these results showcase the benign geometry of bilinear games in the space of measures, enabling particle dynamics with global qualitative convergence guarantees.