# Continuous-time Analysis for Variational Inequalities: An Overview and Desiderata

arXiv: https://arxiv.org/abs/2207.07105 · HTML: https://mlsanity.com/paper/2207.07105/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 50 references that are in the map.
- First posted on arXiv 14 Jul 2022.

## Builds on (most cited first, 25 of 25)

- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [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)
- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (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)
- [A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1702.05575/) (2017)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (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)
- [Last Iterate is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems](https://mlsanity.com/paper/2002.00057/) (2020)
- [The limits of min-max optimization algorithms: convergence to spurious non-critical sets](https://mlsanity.com/paper/2006.09065/) (2020)
- [On Gradient-Based Learning in Continuous Games](https://mlsanity.com/paper/1804.05464/) (2018)
- [A Dynamical Systems Perspective on Nesterov Acceleration](https://mlsanity.com/paper/1905.07436/) (2019)
- [Local Saddle Point Optimization: A Curvature Exploitation Approach](https://mlsanity.com/paper/1805.05751/) (2018)
- [On dissipative symplectic integration with applications to gradient-based optimization](https://mlsanity.com/paper/2004.06840/) (2020)
- [First-order optimization algorithms via inertial systems with Hessian driven damping](https://mlsanity.com/paper/1907.10536/) (2019)
- [ODE Analysis of Stochastic Gradient Methods with Optimism and Anchoring for Minimax Problems](https://mlsanity.com/paper/1905.10899/) (2019)
- [Discretization Drift in Two-Player Games](https://mlsanity.com/paper/2105.13922/) (2021)
- [Multi-Agent Learning in Network Zero-Sum Games is a Hamiltonian System](https://mlsanity.com/paper/1903.01720/) (2019)
- [Fast Optimistic Gradient Descent Ascent (OGDA) method in continuous and discrete time](https://mlsanity.com/paper/2203.10947/) (2022)
- [A Unified Analysis of First-Order Methods for Smooth Games via Integral Quadratic Constraints](https://mlsanity.com/paper/2009.11359/) (2020)
- [Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equations](https://mlsanity.com/paper/2112.13826/) (2021)
- [An O(sr)-Resolution ODE Framework for Understanding Discrete-Time Algorithms and Applications to the Linear Convergence of Minimax Problems](https://mlsanity.com/paper/2001.08826/) (2020)

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

- [SDEs for Minimax Optimization](https://mlsanity.com/paper/2402.12508/) (2024)

## Abstract (arXiv)

Algorithms that solve zero-sum games, multi-objective agent objectives, or, more generally, variational inequality (VI) problems are notoriously unstable on general problems. Owing to the increasing need for solving such problems in machine learning, this instability has been highlighted in recent years as a significant research challenge. In this paper, we provide an overview of recent progress in the use of continuous-time perspectives in the analysis and design of methods targeting the broad VI problem class. Our presentation draws parallels between single-objective problems and multi-objective problems, highlighting the challenges of the latter. We also formulate various desiderata for algorithms that apply to general VIs and we argue that achieving these desiderata may profit from an understanding of the associated continuous-time dynamics.