Accelerating Smooth Games by Manipulating Spectral Shapes
Waïss Azizian, Damien Scieur, Ioannis Mitliagkas, Simon Lacoste-Julien, Gauthier Gidel
Abstract
We use matrix iteration theory to characterize acceleration in smooth games. We define the spectral shape of a family of games as the set containing 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.
BibTeX
@InProceedings{pmlr-v108-azizian20a,
title = {Accelerating Smooth Games by Manipulating Spectral Shapes},
author = {Azizian, Wa\"iss and Scieur, Damien and Mitliagkas, Ioannis and Lacoste-Julien, Simon and Gidel, Gauthier},
booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
pages = {1705--1715},
year = {2020},
editor = {Chiappa, Silvia and Calandra, Roberto},
volume = {108},
series = {Proceedings of Machine Learning Research},
month = {26--28 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v108/azizian20a/azizian20a.pdf},
url = {https://proceedings.mlr.press/v108/azizian20a.html},
abstract = {We use matrix iteration theory to characterize acceleration in smooth games. We define the spectral shape of a family of games as the set containing 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. }
}