SDEs for Minimax Optimization

Enea Monzio Compagnoni, Antonio Orvieto, Hans Kersting, Frank Proske, Aurelien Lucchi
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:4834-4842, 2024.

Abstract

Minimax optimization problems have attracted a lot of attention over the past few years, with applications ranging from economics to machine learning. While advanced optimization methods exist for such problems, characterizing their dynamics in stochastic scenarios remains notably challenging. In this paper, we pioneer the use of stochastic differential equations (SDEs) to analyze and compare Minimax optimizers. Our SDE models for Stochastic Gradient Descent-Ascent, Stochastic Extragradient, and Stochastic Hamiltonian Gradient Descent are provable approximations of their algorithmic counterparts, clearly showcasing the interplay between hyperparameters, implicit regularization, and implicit curvature-induced noise. This perspective also allows for a unified and simplified analysis strategy based on the principles of Itô calculus. Finally, our approach facilitates the derivation of convergence conditions and closed-form solutions for the dynamics in simplified settings, unveiling further insights into the behavior of different optimizers.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-monzio-compagnoni24a, title = { {SDE}s for Minimax Optimization }, author = {Monzio Compagnoni, Enea and Orvieto, Antonio and Kersting, Hans and Proske, Frank and Lucchi, Aurelien}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {4834--4842}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/monzio-compagnoni24a/monzio-compagnoni24a.pdf}, url = {https://proceedings.mlr.press/v238/monzio-compagnoni24a.html}, abstract = { Minimax optimization problems have attracted a lot of attention over the past few years, with applications ranging from economics to machine learning. While advanced optimization methods exist for such problems, characterizing their dynamics in stochastic scenarios remains notably challenging. In this paper, we pioneer the use of stochastic differential equations (SDEs) to analyze and compare Minimax optimizers. Our SDE models for Stochastic Gradient Descent-Ascent, Stochastic Extragradient, and Stochastic Hamiltonian Gradient Descent are provable approximations of their algorithmic counterparts, clearly showcasing the interplay between hyperparameters, implicit regularization, and implicit curvature-induced noise. This perspective also allows for a unified and simplified analysis strategy based on the principles of Itô calculus. Finally, our approach facilitates the derivation of convergence conditions and closed-form solutions for the dynamics in simplified settings, unveiling further insights into the behavior of different optimizers. } }
Endnote
%0 Conference Paper %T SDEs for Minimax Optimization %A Enea Monzio Compagnoni %A Antonio Orvieto %A Hans Kersting %A Frank Proske %A Aurelien Lucchi %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-monzio-compagnoni24a %I PMLR %P 4834--4842 %U https://proceedings.mlr.press/v238/monzio-compagnoni24a.html %V 238 %X Minimax optimization problems have attracted a lot of attention over the past few years, with applications ranging from economics to machine learning. While advanced optimization methods exist for such problems, characterizing their dynamics in stochastic scenarios remains notably challenging. In this paper, we pioneer the use of stochastic differential equations (SDEs) to analyze and compare Minimax optimizers. Our SDE models for Stochastic Gradient Descent-Ascent, Stochastic Extragradient, and Stochastic Hamiltonian Gradient Descent are provable approximations of their algorithmic counterparts, clearly showcasing the interplay between hyperparameters, implicit regularization, and implicit curvature-induced noise. This perspective also allows for a unified and simplified analysis strategy based on the principles of Itô calculus. Finally, our approach facilitates the derivation of convergence conditions and closed-form solutions for the dynamics in simplified settings, unveiling further insights into the behavior of different optimizers.
APA
Monzio Compagnoni, E., Orvieto, A., Kersting, H., Proske, F. & Lucchi, A.. (2024). SDEs for Minimax Optimization . Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:4834-4842 Available from https://proceedings.mlr.press/v238/monzio-compagnoni24a.html.

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