# Fixed-Time Convergence for a Class of Nonconvex-Nonconcave Min-Max Problems

arXiv: https://arxiv.org/abs/2207.12845 · HTML: https://mlsanity.com/paper/2207.12845/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 25 references that are in the map.
- First posted on arXiv 26 Jul 2022.

## Builds on (most cited first, 12 of 12)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Learning Adversarially Fair and Transferable Representations](https://mlsanity.com/paper/1802.06309/) (2018)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Non-convex Min-Max Optimization: Applications, Challenges, and Recent Theoretical Advances](https://mlsanity.com/paper/2006.08141/) (2020)
- [Saddle-point dynamics: conditions for asymptotic stability of saddle points](https://mlsanity.com/paper/1510.02145/) (2015)
- [On the Exponential Stability of Primal-Dual Gradient Dynamics](https://mlsanity.com/paper/1803.01825/) (2018)
- [The proximal augmented Lagrangian method for nonsmooth composite optimization](https://mlsanity.com/paper/1610.04514/) (2016)
- [Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization](https://mlsanity.com/paper/1808.10474/) (2018)
- [Gradient Descent-Ascent Provably Converges to Strict Local Minmax Equilibria with a Finite Timescale Separation](https://mlsanity.com/paper/2009.14820/) (2020)
- [Sampling without Replacement Leads to Faster Rates in Finite-Sum Minimax Optimization](https://mlsanity.com/paper/2206.02953/) (2022)
- [Fixed-Time Stable Proximal Dynamical System for Solving MVIPs](https://mlsanity.com/paper/1908.03517/) (2019)

## Abstract (arXiv)

This study develops a fixed-time convergent saddle point dynamical system for solving min-max problems under a relaxation of standard convexity-concavity assumption. In particular, it is shown that by leveraging the dynamical systems viewpoint of an optimization algorithm, accelerated convergence to a saddle point can be obtained. Instead of requiring the objective function to be strongly-convex--strongly-concave (as necessitated for accelerated convergence of several saddle-point algorithms), uniform fixed-time convergence is guaranteed for functions satisfying only the two-sided Polyak-Lojasiewicz (PL) inequality. A large number of practical problems, including the robust least squares estimation, are known to satisfy the two-sided PL inequality. The proposed method achieves arbitrarily fast convergence compared to any other state-of-the-art method with linear or even super-linear convergence, as also corroborated in numerical case studies.