Provably Optimal Algorithms for Generalized Linear Contextual Bandits

Lihong Li, Yu Lu, Dengyong Zhou
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:2071-2080, 2017.

Abstract

Contextual bandits are widely used in Internet services from news recommendation to advertising, and to Web search. Generalized linear models (logistical regression in particular) have demonstrated stronger performance than linear models in many applications where rewards are binary. However, most theoretical analyses on contextual bandits so far are on linear bandits. In this work, we propose an upper confidence bound based algorithm for generalized linear contextual bandits, which achieves an $\sim O(\sqrt{dT})$ regret over T rounds with d dimensional feature vectors. This regret matches the minimax lower bound, up to logarithmic terms, and improves on the best previous result by a $\sqrt{d}$ factor, assuming the number of arms is fixed. A key component in our analysis is to establish a new, sharp finite-sample confidence bound for maximum likelihood estimates in generalized linear models, which may be of independent interest. We also analyze a simpler upper confidence bound algorithm, which is useful in practice, and prove it to have optimal regret for certain cases.

Cite this Paper

BibTeX
@InProceedings{pmlr-v70-li17c, title = {Provably Optimal Algorithms for Generalized Linear Contextual Bandits}, author = {Lihong Li and Yu Lu and Dengyong Zhou}, booktitle = {Proceedings of the 34th International Conference on Machine Learning}, pages = {2071--2080}, year = {2017}, editor = {Precup, Doina and Teh, Yee Whye}, volume = {70}, series = {Proceedings of Machine Learning Research}, month = {06--11 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v70/li17c/li17c.pdf}, url = {https://proceedings.mlr.press/v70/li17c.html}, abstract = {Contextual bandits are widely used in Internet services from news recommendation to advertising, and to Web search. Generalized linear models (logistical regression in particular) have demonstrated stronger performance than linear models in many applications where rewards are binary. However, most theoretical analyses on contextual bandits so far are on linear bandits. In this work, we propose an upper confidence bound based algorithm for generalized linear contextual bandits, which achieves an $\sim O(\sqrt{dT})$ regret over T rounds with d dimensional feature vectors. This regret matches the minimax lower bound, up to logarithmic terms, and improves on the best previous result by a $\sqrt{d}$ factor, assuming the number of arms is fixed. A key component in our analysis is to establish a new, sharp finite-sample confidence bound for maximum likelihood estimates in generalized linear models, which may be of independent interest. We also analyze a simpler upper confidence bound algorithm, which is useful in practice, and prove it to have optimal regret for certain cases.} }
Endnote
%0 Conference Paper %T Provably Optimal Algorithms for Generalized Linear Contextual Bandits %A Lihong Li %A Yu Lu %A Dengyong Zhou %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-li17c %I PMLR %P 2071--2080 %U https://proceedings.mlr.press/v70/li17c.html %V 70 %X Contextual bandits are widely used in Internet services from news recommendation to advertising, and to Web search. Generalized linear models (logistical regression in particular) have demonstrated stronger performance than linear models in many applications where rewards are binary. However, most theoretical analyses on contextual bandits so far are on linear bandits. In this work, we propose an upper confidence bound based algorithm for generalized linear contextual bandits, which achieves an $\sim O(\sqrt{dT})$ regret over T rounds with d dimensional feature vectors. This regret matches the minimax lower bound, up to logarithmic terms, and improves on the best previous result by a $\sqrt{d}$ factor, assuming the number of arms is fixed. A key component in our analysis is to establish a new, sharp finite-sample confidence bound for maximum likelihood estimates in generalized linear models, which may be of independent interest. We also analyze a simpler upper confidence bound algorithm, which is useful in practice, and prove it to have optimal regret for certain cases.
APA
Li, L., Lu, Y. & Zhou, D.. (2017). Provably Optimal Algorithms for Generalized Linear Contextual Bandits. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:2071-2080 Available from https://proceedings.mlr.press/v70/li17c.html.

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