# On the Stability Connection Between Discrete-Time Algorithms and Their Resolution ODEs: Applications to Min-Max Optimisation

arXiv: https://arxiv.org/abs/2603.01430 · HTML: https://mlsanity.com/paper/2603.01430/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 33 references that are in the map.
- First posted on arXiv 2 Mar 2026.

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

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Deep Decentralized Multi-task Multi-Agent Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/1703.06182/) (2017)
- [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)
- [Last-iterate Convergence of Decentralized Optimistic Gradient Descent/Ascent in Infinite-horizon Competitive Markov Games](https://mlsanity.com/paper/2102.04540/) (2021)
- [The limits of min-max optimization algorithms: convergence to spurious non-critical sets](https://mlsanity.com/paper/2006.09065/) (2020)
- [Stochastic Gradient Descent-Ascent: Unified Theory and New Efficient Methods](https://mlsanity.com/paper/2202.07262/) (2022)
- [Analysis of Optimization Algorithms via Integral Quadratic Constraints: Nonstrongly Convex Problems](https://mlsanity.com/paper/1705.03615/) (2017)
- [Newton-type Methods for Minimax Optimization](https://mlsanity.com/paper/2006.14592/) (2020)
- [Min-Max Optimisation for Nonconvex-Nonconcave Functions Using a Random Zeroth-Order Extragradient Algorithm](https://mlsanity.com/paper/2504.07388/) (2025)
- [Two-timescale Extragradient for Finding Local Minimax Points](https://mlsanity.com/paper/2305.16242/) (2023)
- [Properties of Fixed Points of Generalised Extra Gradient Methods Applied to Min-Max Problems](https://mlsanity.com/paper/2504.03069/) (2025)

## Built on by (most cited first, 1 of 1)

- [From Cursed to Competitive: Closing the ZO-FO Gap via Input-to-State Stability](https://mlsanity.com/paper/2604.25372/) (2026)

## Abstract (arXiv)

This work establishes a rigorous connection between stability properties of discrete-time algorithms (DTAs) and corresponding continuous-time dynamical systems derived through  O(sr) -resolution ordinary differential equations (ODEs). We show that for discrete- and continuous-time dynamical systems satisfying a mild error assumption, exponential stability of a common equilibrium with respect to the continuous time dynamics implies exponential stability of the corresponding equilibrium for the discrete-time dynamics, provided that the step size is chosen sufficiently small. We extend this result to common compact invariant sets. We prove that if an equilibrium is exponentially stable for the  O(sr) -resolution ODE, then it is also exponentially stable for the associated DTA. We apply this framework to analyse the limit point properties of several prominent optimisation algorithms, including Two-Timescale Gradient Descent--Ascent (TT-GDA), Generalised Extragradient (GEG), Two-Timescale Proximal Point (TT-PPM), Damped Newton (DN), Regularised Damped Newton (RDN), and the Jacobian method (JM), by studying their  O(1) - and  O(s) -resolution ODEs. We show that under a proper choice of hyperparameters, the set of saddle points of the objective function is a subset of the set of exponentially stable equilibria of GEG, TT-PPM, DN, and RDN. We relax the common Hessian invariance assumption through direct analysis of the resolution ODEs, broadening the applicability of our results. Numerical examples illustrate the theoretical findings.