Quantum Speedups for Zero-Sum Games via Improved Dynamic Gibbs Sampling

Adam Bouland, Yosheb M Getachew, Yujia Jin, Aaron Sidford, Kevin Tian
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:2932-2952, 2023.

Abstract

We give a quantum algorithm for computing an $\epsilon$-approximate Nash equilibrium of a zero-sum game in a $m \times n$ payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time $\widetilde{O}(\sqrt{m + n}\cdot \epsilon^{-2.5} + \epsilon^{-3})$ and outputs a classical representation of the $\epsilon$-approximate Nash equilibrium. This improves upon the best prior quantum runtime of $\widetilde{O}(\sqrt{m + n} \cdot \epsilon^{-3})$ obtained by [van Apeldoorn, Gilyen ’19] and the classical $\widetilde{O}((m + n) \cdot \epsilon^{-2})$ runtime due to [Grigoradis, Khachiyan ’95] whenever $\epsilon = \Omega((m +n)^{-1})$. We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-bouland23a, title = {Quantum Speedups for Zero-Sum Games via Improved Dynamic {G}ibbs Sampling}, author = {Bouland, Adam and Getachew, Yosheb M and Jin, Yujia and Sidford, Aaron and Tian, Kevin}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {2932--2952}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/bouland23a/bouland23a.pdf}, url = {https://proceedings.mlr.press/v202/bouland23a.html}, abstract = {We give a quantum algorithm for computing an $\epsilon$-approximate Nash equilibrium of a zero-sum game in a $m \times n$ payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time $\widetilde{O}(\sqrt{m + n}\cdot \epsilon^{-2.5} + \epsilon^{-3})$ and outputs a classical representation of the $\epsilon$-approximate Nash equilibrium. This improves upon the best prior quantum runtime of $\widetilde{O}(\sqrt{m + n} \cdot \epsilon^{-3})$ obtained by [van Apeldoorn, Gilyen ’19] and the classical $\widetilde{O}((m + n) \cdot \epsilon^{-2})$ runtime due to [Grigoradis, Khachiyan ’95] whenever $\epsilon = \Omega((m +n)^{-1})$. We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.} }
Endnote
%0 Conference Paper %T Quantum Speedups for Zero-Sum Games via Improved Dynamic Gibbs Sampling %A Adam Bouland %A Yosheb M Getachew %A Yujia Jin %A Aaron Sidford %A Kevin Tian %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-bouland23a %I PMLR %P 2932--2952 %U https://proceedings.mlr.press/v202/bouland23a.html %V 202 %X We give a quantum algorithm for computing an $\epsilon$-approximate Nash equilibrium of a zero-sum game in a $m \times n$ payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time $\widetilde{O}(\sqrt{m + n}\cdot \epsilon^{-2.5} + \epsilon^{-3})$ and outputs a classical representation of the $\epsilon$-approximate Nash equilibrium. This improves upon the best prior quantum runtime of $\widetilde{O}(\sqrt{m + n} \cdot \epsilon^{-3})$ obtained by [van Apeldoorn, Gilyen ’19] and the classical $\widetilde{O}((m + n) \cdot \epsilon^{-2})$ runtime due to [Grigoradis, Khachiyan ’95] whenever $\epsilon = \Omega((m +n)^{-1})$. We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.
APA
Bouland, A., Getachew, Y.M., Jin, Y., Sidford, A. & Tian, K.. (2023). Quantum Speedups for Zero-Sum Games via Improved Dynamic Gibbs Sampling. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:2932-2952 Available from https://proceedings.mlr.press/v202/bouland23a.html.

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