# Linear Lower Bounds and Conditioning of Differentiable Games

arXiv: https://arxiv.org/abs/1906.07300 · HTML: https://mlsanity.com/paper/1906.07300/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2019.

## Built on by (most cited first, 3 of 3)

- [Convex-Concave Min-Max Stackelberg Games](https://mlsanity.com/paper/2110.05192/) (2021)
- [On Convergence of Gradient Descent Ascent: A Tight Local Analysis](https://mlsanity.com/paper/2207.00957/) (2022)
- [Robust No-Regret Learning in Min-Max Stackelberg Games](https://mlsanity.com/paper/2203.14126/) (2022)

## Abstract (arXiv)

Recent successes of game-theoretic formulations in ML have caused a resurgence of research interest in differentiable games. Overwhelmingly, that research focuses on methods and upper bounds on their speed of convergence. In this work, we approach the question of fundamental iteration complexity by providing lower bounds to complement the linear (i.e. geometric) upper bounds observed in the literature on a wide class of problems. We cast saddle-point and min-max problems as 2-player games. We leverage tools from single-objective convex optimisation to propose new linear lower bounds for convex-concave games. Notably, we give a linear lower bound for n-player differentiable games, by using the spectral properties of the update operator. We then propose a new definition of the condition number arising from our lower bound analysis. Unlike past definitions, our condition number captures the fact that linear rates are possible in games, even in the absence of strong convexity or strong concavity in the variables.