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  1. Banerjee, Arindam ; Fukumizu, Kenji (Ed.)
    We study a constrained contextual linear bandit setting, where the goal of the agent is to produce a sequence of policies, whose expected cumulative reward over the course of multiple rounds is maximum, and each one of them has an expected cost below a certain threshold. We propose an upper-confidence bound algorithm for this problem, called optimistic pessimistic linear bandit (OPLB), and prove a sublinear bound on its regret that is inversely proportional to the difference between the constraint threshold and the cost of a known feasible action. Our algorithm balances exploration and constraint satisfaction using a novel idea thatmore »scales the radii of the reward and cost confidence sets with different scaling factors. We further specialize our results to multi-armed bandits and propose a computationally efficient algorithm for this setting and prove a a regret bound that is better than simply casting multi-armed bandits as an instance of linear bandits and using the regret bound of OPLB. We also prove a lower-bound for the problem studied in the paper and provide simulations to validate our theoretical results. Finally, we show how our algorithm and analysis can be extended to multiple constraints and to the case when the cost of the feasible action is unknown.« less
  2. Daumé III, Hal ; Singh, Aarti (Ed.)
    Thompson sampling for multi-armed bandit problems is known to enjoy favorable performance in both theory and practice. However, its wider deployment is restricted due to a significant computational limitation: the need for samples from posterior distributions at every iteration. In practice, this limitation is alleviated by making use of approximate sampling methods, yet provably incorporating approximate samples into Thompson Sampling algorithms remains an open problem. In this work we address this by proposing two efficient Langevin MCMC algorithms tailored to Thompson sampling. The resulting approximate Thompson Sampling algorithms are efficiently implementable and provably achieve optimal instance-dependent regret for the Multi-Armedmore »Bandit (MAB) problem. To prove these results we derive novel posterior concentration bounds and MCMC convergence rates for log-concave distributions which may be of independent interest.« less