Gauthier Gidel

We consider diﬀerentiable games where the goal is to ﬁnd a Nash equilibrium. The machine learning community has recently started using v… (voir plus)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

Negative Momentum for Improved Game Dynamics

Reyhane Askari Hemmat

Mohammad Pezeshki

Gabriel Huang

Rémi LE PRIOL

Ioannis Mitliagkas

Games generalize the single-objective optimization paradigm by introducing different objective functions for different players. Differentiab… (voir plus)le games often proceed by simultaneous or alternating gradient updates. In machine learning, games are gaining new importance through formulations like generative adversarial networks (GANs) and actor-critic systems. However, compared to single-objective optimization, game dynamics are more complex and less understood. In this paper, we analyze gradient-based methods with momentum on simple games. We prove that alternating updates are more stable than simultaneous updates. Next, we show both theoretically and empirically that alternating gradient updates with a negative momentum term achieves convergence in a difficult toy adversarial problem, but also on the notoriously difficult to train saturating GANs.

2019-04-11

Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics (publié)

proceedings.mlr.press

Negative Momentum for Improved Game Dynamics

Reyhane Askari Hemmat

Mohammad Pezeshki

Gabriel Huang

Rémi LE PRIOL

Ioannis Mitliagkas

2018-07-12

ArXiv (preprint)

Frank-Wolfe Splitting via Augmented Lagrangian Method

Fabian Pedregosa

Minimizing a function over an intersection of convex sets is an important task in optimization that is often much more challenging than mini… (voir plus)mizing it over each individual constraint set. While traditional methods such as Frank-Wolfe (FW) or proximal gradient descent assume access to a linear or quadratic oracle on the intersection, splitting techniques take advantage of the structure of each sets, and only require access to the oracle on the individual constraints. In this work, we develop and analyze the Frank-Wolfe Augmented Lagrangian (FW-AL) algorithm, a method for minimizing a smooth function over convex compact sets related by a "linear consistency" constraint that only requires access to a linear minimization oracle over the individual constraints. It is based on the Augmented Lagrangian Method (ALM), also known as Method of Multipliers, but unlike most existing splitting methods, it only requires access to linear (instead of quadratic) minimization oracles. We use recent advances in the analysis of Frank-Wolfe and the alternating direction method of multipliers algorithms to prove a sublinear convergence rate for FW-AL over general convex compact sets and a linear convergence rate for polytopes.

2018-03-31

ArXiv (preprint)

Frank-Wolfe Splitting via Augmented Lagrangian Method

Fabian Pedregosa

2018-01-01

AISTATS (publié)

proceedings.mlr.press