Infinite-Dimensional Optimization for Zero-Sum Games via Variational Transport

Lewis Liu, Yufeng Zhang, Zhuoran Yang, Reza Babanezhad, Zhaoran Wang
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:7033-7044, 2021.

Abstract

Game optimization has been extensively studied when decision variables lie in a finite-dimensional space, of which solutions correspond to pure strategies at the Nash equilibrium (NE), and the gradient descent-ascent (GDA) method works widely in practice. In this paper, we consider infinite-dimensional zero-sum games by a min-max distributional optimization problem over a space of probability measures defined on a continuous variable set, which is inspired by finding a mixed NE for finite-dimensional zero-sum games. We then aim to answer the following question: \textit{Will GDA-type algorithms still be provably efficient when extended to infinite-dimensional zero-sum games?} To answer this question, we propose a particle-based variational transport algorithm based on GDA in the functional spaces. Specifically, the algorithm performs multi-step functional gradient descent-ascent in the Wasserstein space via pushing two sets of particles in the variable space. By characterizing the gradient estimation error from variational form maximization and the convergence behavior of each player with different objective landscapes, we prove rigorously that the generalized GDA algorithm converges to the NE or the value of the game efficiently for a class of games under the Polyak-Ł{ojasiewicz} (PL) condition. To conclude, we provide complete statistical and convergence guarantees for solving an infinite-dimensional zero-sum game via a provably efficient particle-based method. Additionally, our work provides the first thorough statistical analysis for the particle-based algorithm to learn an objective functional with a variational form using universal approximators (\textit{i.e.}, neural networks (NNs)), which is of independent interest.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-liu21ac, title = {Infinite-Dimensional Optimization for Zero-Sum Games via Variational Transport}, author = {Liu, Lewis and Zhang, Yufeng and Yang, Zhuoran and Babanezhad, Reza and Wang, Zhaoran}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {7033--7044}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/liu21ac/liu21ac.pdf}, url = {https://proceedings.mlr.press/v139/liu21ac.html}, abstract = {Game optimization has been extensively studied when decision variables lie in a finite-dimensional space, of which solutions correspond to pure strategies at the Nash equilibrium (NE), and the gradient descent-ascent (GDA) method works widely in practice. In this paper, we consider infinite-dimensional zero-sum games by a min-max distributional optimization problem over a space of probability measures defined on a continuous variable set, which is inspired by finding a mixed NE for finite-dimensional zero-sum games. We then aim to answer the following question: \textit{Will GDA-type algorithms still be provably efficient when extended to infinite-dimensional zero-sum games?} To answer this question, we propose a particle-based variational transport algorithm based on GDA in the functional spaces. Specifically, the algorithm performs multi-step functional gradient descent-ascent in the Wasserstein space via pushing two sets of particles in the variable space. By characterizing the gradient estimation error from variational form maximization and the convergence behavior of each player with different objective landscapes, we prove rigorously that the generalized GDA algorithm converges to the NE or the value of the game efficiently for a class of games under the Polyak-Ł{ojasiewicz} (PL) condition. To conclude, we provide complete statistical and convergence guarantees for solving an infinite-dimensional zero-sum game via a provably efficient particle-based method. Additionally, our work provides the first thorough statistical analysis for the particle-based algorithm to learn an objective functional with a variational form using universal approximators (\textit{i.e.}, neural networks (NNs)), which is of independent interest.} }
Endnote
%0 Conference Paper %T Infinite-Dimensional Optimization for Zero-Sum Games via Variational Transport %A Lewis Liu %A Yufeng Zhang %A Zhuoran Yang %A Reza Babanezhad %A Zhaoran Wang %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-liu21ac %I PMLR %P 7033--7044 %U https://proceedings.mlr.press/v139/liu21ac.html %V 139 %X Game optimization has been extensively studied when decision variables lie in a finite-dimensional space, of which solutions correspond to pure strategies at the Nash equilibrium (NE), and the gradient descent-ascent (GDA) method works widely in practice. In this paper, we consider infinite-dimensional zero-sum games by a min-max distributional optimization problem over a space of probability measures defined on a continuous variable set, which is inspired by finding a mixed NE for finite-dimensional zero-sum games. We then aim to answer the following question: \textit{Will GDA-type algorithms still be provably efficient when extended to infinite-dimensional zero-sum games?} To answer this question, we propose a particle-based variational transport algorithm based on GDA in the functional spaces. Specifically, the algorithm performs multi-step functional gradient descent-ascent in the Wasserstein space via pushing two sets of particles in the variable space. By characterizing the gradient estimation error from variational form maximization and the convergence behavior of each player with different objective landscapes, we prove rigorously that the generalized GDA algorithm converges to the NE or the value of the game efficiently for a class of games under the Polyak-Ł{ojasiewicz} (PL) condition. To conclude, we provide complete statistical and convergence guarantees for solving an infinite-dimensional zero-sum game via a provably efficient particle-based method. Additionally, our work provides the first thorough statistical analysis for the particle-based algorithm to learn an objective functional with a variational form using universal approximators (\textit{i.e.}, neural networks (NNs)), which is of independent interest.
APA
Liu, L., Zhang, Y., Yang, Z., Babanezhad, R. & Wang, Z.. (2021). Infinite-Dimensional Optimization for Zero-Sum Games via Variational Transport. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:7033-7044 Available from https://proceedings.mlr.press/v139/liu21ac.html.

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