A Convergent and Dimension-Independent Min-Max Optimization Algorithm

Vijay Keswani, Oren Mangoubi, Sushant Sachdeva, Nisheeth K. Vishnoi
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:10939-10973, 2022.

Abstract

We study a variant of a recently introduced min-max optimization framework where the max-player is constrained to update its parameters in a greedy manner until it reaches a first-order stationary point. Our equilibrium definition for this framework depends on a proposal distribution which the min-player uses to choose directions in which to update its parameters. We show that, given a smooth and bounded nonconvex-nonconcave objective function, access to any proposal distribution for the min-player’s updates, and stochastic gradient oracle for the max-player, our algorithm converges to the aforementioned approximate local equilibrium in a number of iterations that does not depend on the dimension. The equilibrium point found by our algorithm depends on the proposal distribution, and when applying our algorithm to train GANs we choose the proposal distribution to be a distribution of stochastic gradients. We empirically evaluate our algorithm on challenging nonconvex-nonconcave test-functions and loss functions arising in GAN training. Our algorithm converges on these test functions and, when used to train GANs, trains stably on synthetic and real-world datasets and avoids mode collapse.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-keswani22a, title = {A Convergent and Dimension-Independent Min-Max Optimization Algorithm}, author = {Keswani, Vijay and Mangoubi, Oren and Sachdeva, Sushant and Vishnoi, Nisheeth K.}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {10939--10973}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/keswani22a/keswani22a.pdf}, url = {https://proceedings.mlr.press/v162/keswani22a.html}, abstract = {We study a variant of a recently introduced min-max optimization framework where the max-player is constrained to update its parameters in a greedy manner until it reaches a first-order stationary point. Our equilibrium definition for this framework depends on a proposal distribution which the min-player uses to choose directions in which to update its parameters. We show that, given a smooth and bounded nonconvex-nonconcave objective function, access to any proposal distribution for the min-player’s updates, and stochastic gradient oracle for the max-player, our algorithm converges to the aforementioned approximate local equilibrium in a number of iterations that does not depend on the dimension. The equilibrium point found by our algorithm depends on the proposal distribution, and when applying our algorithm to train GANs we choose the proposal distribution to be a distribution of stochastic gradients. We empirically evaluate our algorithm on challenging nonconvex-nonconcave test-functions and loss functions arising in GAN training. Our algorithm converges on these test functions and, when used to train GANs, trains stably on synthetic and real-world datasets and avoids mode collapse.} }
Endnote
%0 Conference Paper %T A Convergent and Dimension-Independent Min-Max Optimization Algorithm %A Vijay Keswani %A Oren Mangoubi %A Sushant Sachdeva %A Nisheeth K. Vishnoi %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-keswani22a %I PMLR %P 10939--10973 %U https://proceedings.mlr.press/v162/keswani22a.html %V 162 %X We study a variant of a recently introduced min-max optimization framework where the max-player is constrained to update its parameters in a greedy manner until it reaches a first-order stationary point. Our equilibrium definition for this framework depends on a proposal distribution which the min-player uses to choose directions in which to update its parameters. We show that, given a smooth and bounded nonconvex-nonconcave objective function, access to any proposal distribution for the min-player’s updates, and stochastic gradient oracle for the max-player, our algorithm converges to the aforementioned approximate local equilibrium in a number of iterations that does not depend on the dimension. The equilibrium point found by our algorithm depends on the proposal distribution, and when applying our algorithm to train GANs we choose the proposal distribution to be a distribution of stochastic gradients. We empirically evaluate our algorithm on challenging nonconvex-nonconcave test-functions and loss functions arising in GAN training. Our algorithm converges on these test functions and, when used to train GANs, trains stably on synthetic and real-world datasets and avoids mode collapse.
APA
Keswani, V., Mangoubi, O., Sachdeva, S. & Vishnoi, N.K.. (2022). A Convergent and Dimension-Independent Min-Max Optimization Algorithm. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:10939-10973 Available from https://proceedings.mlr.press/v162/keswani22a.html.

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