Thompson Sampling for Linearly Constrained Bandits

• URL: http://arxiv.org/abs/2004.09258v2
• Date: Tue, 12 May 2020 18:34:47 GMT
• Title: Thompson Sampling for Linearly Constrained Bandits
• Authors: Vidit Saxena, Joseph E. Gonzalez, and Joakim Jald\'en
• Abstract summary: We describe LinConTS, an algorithm for bandits that place a linear constraint on the probability of earning a reward in every round. We show that for LinConTS, the regret as well as the cumulative constraint violations are upper bounded by O(log T) for the suboptimal arms.
• Score: 18.477962150017095
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: We address multi-armed bandits (MAB) where the objective is to maximize the cumulative reward under a probabilistic linear constraint. For a few real-world instances of this problem, constrained extensions of the well-known Thompson Sampling (TS) heuristic have recently been proposed. However, finite-time analysis of constrained TS is challenging; as a result, only O(\sqrt{T}) bounds on the cumulative reward loss (i.e., the regret) are available. In this paper, we describe LinConTS, a TS-based algorithm for bandits that place a linear constraint on the probability of earning a reward in every round. We show that for LinConTS, the regret as well as the cumulative constraint violations are upper bounded by O(\log T) for the suboptimal arms. We develop a proof technique that relies on careful analysis of the dual problem and combine it with recent theoretical work on unconstrained TS. Through numerical experiments on two real-world datasets, we demonstrate that LinConTS outperforms an asymptotically optimal upper confidence bound (UCB) scheme in terms of simultaneously minimizing the regret and the violation.

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