Nearly Minimax-Optimal Regret for Linearly Parameterized Bandits

Yingkai Li, Yining Wang, Yuan Zhou
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:2173-2174, 2019.

Abstract

We study the linear contextual bandit problem with finite action sets. When the problem dimension is $d$, the time horizon is $T$, and there are $n \leq 2^{d/2}$ candidate actions per time period, we (1) show that the minimax expected regret is $\Omega(\sqrt{dT \log T \log n})$ for every algorithm, and (2) introduce a Variable-Confidence-Level (VCL) SupLinUCB algorithm whose regret matches the lower bound up to iterated logarithmic factors. Our algorithmic result saves two $\sqrt{\log T}$ factors from previous analysis, and our information-theoretical lower bound also improves previous results by one $\sqrt{\log T}$ factor, revealing a regret scaling quite different from classical multi-armed bandits in which no logarithmic $T$ term is present in minimax regret. Our proof techniques include variable confidence levels and a careful analysis of layer sizes of SupLinUCB on the upper bound side, and delicately constructed adversarial sequences showing the tightness of elliptical potential lemmas on the lower bound side.

Cite this Paper

BibTeX
@InProceedings{pmlr-v99-li19b, title = {Nearly Minimax-Optimal Regret for Linearly Parameterized Bandits}, author = {Li, Yingkai and Wang, Yining and Zhou, Yuan}, booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, pages = {2173--2174}, year = {2019}, editor = {Beygelzimer, Alina and Hsu, Daniel}, volume = {99}, series = {Proceedings of Machine Learning Research}, month = {25--28 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v99/li19b/li19b.pdf}, url = {https://proceedings.mlr.press/v99/li19b.html}, abstract = {We study the linear contextual bandit problem with finite action sets. When the problem dimension is $d$, the time horizon is $T$, and there are $n \leq 2^{d/2}$ candidate actions per time period, we (1) show that the minimax expected regret is $\Omega(\sqrt{dT \log T \log n})$ for every algorithm, and (2) introduce a Variable-Confidence-Level (VCL) SupLinUCB algorithm whose regret matches the lower bound up to iterated logarithmic factors. Our algorithmic result saves two $\sqrt{\log T}$ factors from previous analysis, and our information-theoretical lower bound also improves previous results by one $\sqrt{\log T}$ factor, revealing a regret scaling quite different from classical multi-armed bandits in which no logarithmic $T$ term is present in minimax regret. Our proof techniques include variable confidence levels and a careful analysis of layer sizes of SupLinUCB on the upper bound side, and delicately constructed adversarial sequences showing the tightness of elliptical potential lemmas on the lower bound side. } }
Endnote
%0 Conference Paper %T Nearly Minimax-Optimal Regret for Linearly Parameterized Bandits %A Yingkai Li %A Yining Wang %A Yuan Zhou %B Proceedings of the Thirty-Second Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Alina Beygelzimer %E Daniel Hsu %F pmlr-v99-li19b %I PMLR %P 2173--2174 %U https://proceedings.mlr.press/v99/li19b.html %V 99 %X We study the linear contextual bandit problem with finite action sets. When the problem dimension is $d$, the time horizon is $T$, and there are $n \leq 2^{d/2}$ candidate actions per time period, we (1) show that the minimax expected regret is $\Omega(\sqrt{dT \log T \log n})$ for every algorithm, and (2) introduce a Variable-Confidence-Level (VCL) SupLinUCB algorithm whose regret matches the lower bound up to iterated logarithmic factors. Our algorithmic result saves two $\sqrt{\log T}$ factors from previous analysis, and our information-theoretical lower bound also improves previous results by one $\sqrt{\log T}$ factor, revealing a regret scaling quite different from classical multi-armed bandits in which no logarithmic $T$ term is present in minimax regret. Our proof techniques include variable confidence levels and a careful analysis of layer sizes of SupLinUCB on the upper bound side, and delicately constructed adversarial sequences showing the tightness of elliptical potential lemmas on the lower bound side.
APA
Li, Y., Wang, Y. & Zhou, Y.. (2019). Nearly Minimax-Optimal Regret for Linearly Parameterized Bandits. Proceedings of the Thirty-Second Conference on Learning Theory, in Proceedings of Machine Learning Research 99:2173-2174 Available from https://proceedings.mlr.press/v99/li19b.html.

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