Smoothed Adversarial Linear Contextual Bandits with Knapsacks

Vidyashankar Sivakumar, Shiliang Zuo, Arindam Banerjee
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:20253-20277, 2022.

Abstract

Many bandit problems are characterized by the learner making decisions under constraints. The learner in Linear Contextual Bandits with Knapsacks (LinCBwK) receives a resource consumption vector in addition to a scalar reward in each time step which are both linear functions of the context corresponding to the chosen arm. For a fixed time horizon $T$, the goal of the learner is to maximize rewards while ensuring resource consumptions do not exceed a pre-specified budget. We present algorithms and characterize regret for LinCBwK in the smoothed setting where base context vectors are assumed to be perturbed by Gaussian noise. We consider both the stochastic and adversarial settings for the base contexts, and our analysis of stochastic LinCBwK can be viewed as a warm-up to the more challenging adversarial LinCBwK. For the stochastic setting, we obtain $O(\sqrt{T})$ additive regret bounds compared to the best context dependent fixed policy. The analysis combines ideas for greedy parameter estimation in \cite{kmrw18, siwb20} and the primal-dual paradigm first explored in \cite{agde17, agde14}. Our main contribution is an algorithm with $O(\log T)$ competitive ratio relative to the best context dependent fixed policy for the adversarial setting. The algorithm for the adversarial setting employs ideas from the primal-dual framework \cite{agde17, agde14} and a novel adaptation of the doubling trick \cite{isss19}.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-sivakumar22a, title = {Smoothed Adversarial Linear Contextual Bandits with Knapsacks}, author = {Sivakumar, Vidyashankar and Zuo, Shiliang and Banerjee, Arindam}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {20253--20277}, 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/sivakumar22a/sivakumar22a.pdf}, url = {https://proceedings.mlr.press/v162/sivakumar22a.html}, abstract = {Many bandit problems are characterized by the learner making decisions under constraints. The learner in Linear Contextual Bandits with Knapsacks (LinCBwK) receives a resource consumption vector in addition to a scalar reward in each time step which are both linear functions of the context corresponding to the chosen arm. For a fixed time horizon $T$, the goal of the learner is to maximize rewards while ensuring resource consumptions do not exceed a pre-specified budget. We present algorithms and characterize regret for LinCBwK in the smoothed setting where base context vectors are assumed to be perturbed by Gaussian noise. We consider both the stochastic and adversarial settings for the base contexts, and our analysis of stochastic LinCBwK can be viewed as a warm-up to the more challenging adversarial LinCBwK. For the stochastic setting, we obtain $O(\sqrt{T})$ additive regret bounds compared to the best context dependent fixed policy. The analysis combines ideas for greedy parameter estimation in \cite{kmrw18, siwb20} and the primal-dual paradigm first explored in \cite{agde17, agde14}. Our main contribution is an algorithm with $O(\log T)$ competitive ratio relative to the best context dependent fixed policy for the adversarial setting. The algorithm for the adversarial setting employs ideas from the primal-dual framework \cite{agde17, agde14} and a novel adaptation of the doubling trick \cite{isss19}.} }
Endnote
%0 Conference Paper %T Smoothed Adversarial Linear Contextual Bandits with Knapsacks %A Vidyashankar Sivakumar %A Shiliang Zuo %A Arindam Banerjee %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-sivakumar22a %I PMLR %P 20253--20277 %U https://proceedings.mlr.press/v162/sivakumar22a.html %V 162 %X Many bandit problems are characterized by the learner making decisions under constraints. The learner in Linear Contextual Bandits with Knapsacks (LinCBwK) receives a resource consumption vector in addition to a scalar reward in each time step which are both linear functions of the context corresponding to the chosen arm. For a fixed time horizon $T$, the goal of the learner is to maximize rewards while ensuring resource consumptions do not exceed a pre-specified budget. We present algorithms and characterize regret for LinCBwK in the smoothed setting where base context vectors are assumed to be perturbed by Gaussian noise. We consider both the stochastic and adversarial settings for the base contexts, and our analysis of stochastic LinCBwK can be viewed as a warm-up to the more challenging adversarial LinCBwK. For the stochastic setting, we obtain $O(\sqrt{T})$ additive regret bounds compared to the best context dependent fixed policy. The analysis combines ideas for greedy parameter estimation in \cite{kmrw18, siwb20} and the primal-dual paradigm first explored in \cite{agde17, agde14}. Our main contribution is an algorithm with $O(\log T)$ competitive ratio relative to the best context dependent fixed policy for the adversarial setting. The algorithm for the adversarial setting employs ideas from the primal-dual framework \cite{agde17, agde14} and a novel adaptation of the doubling trick \cite{isss19}.
APA
Sivakumar, V., Zuo, S. & Banerjee, A.. (2022). Smoothed Adversarial Linear Contextual Bandits with Knapsacks. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:20253-20277 Available from https://proceedings.mlr.press/v162/sivakumar22a.html.

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