Restless bandits with rewards generated by a linear Gaussian dynamical system

Jonathan Gornet, Bruno Sinopoli
Proceedings of the 6th Annual Learning for Dynamics & Control Conference, PMLR 242:1791-1802, 2024.

Abstract

Decision-making under uncertainty is a fundamental problem encountered frequently and can be formulated as a stochastic multi-armed bandit problem. In the problem, the learner interacts with an environment by choosing an action at each round, where a round is an instance of an interaction. In response, the environment reveals a reward, which is sampled from a stochastic process, to the learner. The goal of the learner is to maximize cumulative reward. In this work, we assume that the rewards are the inner product of an action vector and a state vector generated by a linear Gaussian dynamical system. To predict the reward for each action, we propose a method that takes a linear combination of previously observed rewards for predicting each action’s next reward. We show that, regardless of the sequence of previous actions chosen, the reward sampled for any previously chosen action can be used for predicting another action’s future reward, i.e. the reward sampled for action 1 at round $t-1$ can be used for predicting the reward for action $2$ at round $t$. This is accomplished by designing a modified Kalman filter with a matrix representation that can be learned for reward prediction. Numerical evaluations are carried out on a set of linear Gaussian dynamical systems and are compared with 2 other well-known stochastic multi-armed bandit algorithms.

Cite this Paper

BibTeX
@InProceedings{pmlr-v242-gornet24a, title = {Restless bandits with rewards generated by a linear {G}aussian dynamical system}, author = {Gornet, Jonathan and Sinopoli, Bruno}, booktitle = {Proceedings of the 6th Annual Learning for Dynamics & Control Conference}, pages = {1791--1802}, year = {2024}, editor = {Abate, Alessandro and Cannon, Mark and Margellos, Kostas and Papachristodoulou, Antonis}, volume = {242}, series = {Proceedings of Machine Learning Research}, month = {15--17 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v242/gornet24a/gornet24a.pdf}, url = {https://proceedings.mlr.press/v242/gornet24a.html}, abstract = {Decision-making under uncertainty is a fundamental problem encountered frequently and can be formulated as a stochastic multi-armed bandit problem. In the problem, the learner interacts with an environment by choosing an action at each round, where a round is an instance of an interaction. In response, the environment reveals a reward, which is sampled from a stochastic process, to the learner. The goal of the learner is to maximize cumulative reward. In this work, we assume that the rewards are the inner product of an action vector and a state vector generated by a linear Gaussian dynamical system. To predict the reward for each action, we propose a method that takes a linear combination of previously observed rewards for predicting each action’s next reward. We show that, regardless of the sequence of previous actions chosen, the reward sampled for any previously chosen action can be used for predicting another action’s future reward, i.e. the reward sampled for action 1 at round $t-1$ can be used for predicting the reward for action $2$ at round $t$. This is accomplished by designing a modified Kalman filter with a matrix representation that can be learned for reward prediction. Numerical evaluations are carried out on a set of linear Gaussian dynamical systems and are compared with 2 other well-known stochastic multi-armed bandit algorithms.} }
Endnote
%0 Conference Paper %T Restless bandits with rewards generated by a linear Gaussian dynamical system %A Jonathan Gornet %A Bruno Sinopoli %B Proceedings of the 6th Annual Learning for Dynamics & Control Conference %C Proceedings of Machine Learning Research %D 2024 %E Alessandro Abate %E Mark Cannon %E Kostas Margellos %E Antonis Papachristodoulou %F pmlr-v242-gornet24a %I PMLR %P 1791--1802 %U https://proceedings.mlr.press/v242/gornet24a.html %V 242 %X Decision-making under uncertainty is a fundamental problem encountered frequently and can be formulated as a stochastic multi-armed bandit problem. In the problem, the learner interacts with an environment by choosing an action at each round, where a round is an instance of an interaction. In response, the environment reveals a reward, which is sampled from a stochastic process, to the learner. The goal of the learner is to maximize cumulative reward. In this work, we assume that the rewards are the inner product of an action vector and a state vector generated by a linear Gaussian dynamical system. To predict the reward for each action, we propose a method that takes a linear combination of previously observed rewards for predicting each action’s next reward. We show that, regardless of the sequence of previous actions chosen, the reward sampled for any previously chosen action can be used for predicting another action’s future reward, i.e. the reward sampled for action 1 at round $t-1$ can be used for predicting the reward for action $2$ at round $t$. This is accomplished by designing a modified Kalman filter with a matrix representation that can be learned for reward prediction. Numerical evaluations are carried out on a set of linear Gaussian dynamical systems and are compared with 2 other well-known stochastic multi-armed bandit algorithms.
APA
Gornet, J. & Sinopoli, B.. (2024). Restless bandits with rewards generated by a linear Gaussian dynamical system. Proceedings of the 6th Annual Learning for Dynamics & Control Conference, in Proceedings of Machine Learning Research 242:1791-1802 Available from https://proceedings.mlr.press/v242/gornet24a.html.

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