Waïss Azizian

Linear Lower Bounds and Conditioning of Differentiable Games

Recent successes of game-theoretic formulations in ML have caused a resurgence of research interest in differentiable games. Overwhelmingly,… (see more) 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

2020-11-21

Proceedings of the 37th International Conference on Machine Learning (published)

proceedings.mlr.press

Accelerating Smooth Games by Manipulating Spectral Shapes

2020-06-03

Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics (published)

proceedings.mlr.press

arxiv.org

Accelerating Smooth Games by Manipulating Spectral Shapes

We use matrix iteration theory to characterize acceleration in smooth games. We define the spectral shape of a family of games as the set co… (see more)ntaining all eigenvalues of the Jacobians of standard gradient dynamics in the family. Shapes restricted to the real line represent well-understood classes of problems, like minimization. Shapes spanning the complex plane capture the added numerical challenges in solving smooth games. In this framework, we describe gradient-based methods, such as extragradient, as transformations on the spectral shape. Using this perspective, we propose an optimal algorithm for bilinear games. For smooth and strongly monotone operators, we identify a continuum between convex minimization, where acceleration is possible using Polyak's momentum, and the worst case where gradient descent is optimal. Finally, going beyond first-order methods, we propose an accelerated version of consensus optimization.

2020-01-02

ArXiv (preprint)

arxiv.org

A Tight and Unified Analysis of Gradient-Based Methods for a Whole Spectrum of Differentiable Games

We consider diﬀerentiable games where the goal is to ﬁnd a Nash equilibrium. The machine learning community has recently started using v… (see more)ariants of the gradient method ( GD ). Prime examples are extragradient ( EG ), the optimistic gradient method ( OG ) and consensus optimization ( CO ), which enjoy linear convergence in cases like bilinear games, where the standard GD fails. The full bene-ﬁts of theses relatively new methods are not known as there is no uniﬁed analysis for both strongly monotone and bilinear games. We provide new analyses of the EG ’s local and global convergence properties and use is to get a tighter global convergence rate for OG and CO . Our analysis covers the whole range of settings between bilinear and strongly monotone games. It reveals that these methods converges via diﬀerent mechanisms at these extremes; in between, it exploits the most favorable mechanism for the given problem. We then prove that EG achieves the optimal rate for a wide class of algorithms with any number of extrapolations. Our tight analysis of EG ’s convergence rate in games shows that, unlike in convex minimization, EG may be much faster than GD .

2020-01-01

International Conference on Artificial Intelligence and Statistics (published)

dblp.uni-trier.de

A Tight and Unified Analysis of Gradient-Based Methods for a Whole Spectrum of Differentiable Games.

2020-01-01

International Conference on Artificial Intelligence and Statistics (published)

proceedings.mlr.press