Global Bandits

TL;DR

This paper introduces 'global bandits', a new class of multi-armed bandit models where rewards are correlated through a single unknown parameter, and proposes algorithms with bounded and sub-linear regret for such models.

Contribution

The paper defines global bandits with correlated rewards, and develops algorithms that achieve bounded and sub-linear regret, improving decision-making in correlated reward settings.

Findings

The greedy policy achieves bounded regret depending on the true parameter.

A variant attains $ ilde{O}( oot{T}{})$ worst-case regret.

Experiments show significant gains in dynamic pricing applications.

Abstract

Multi-armed bandits (MAB) model sequential decision making problems, in which a learner sequentially chooses arms with unknown reward distributions in order to maximize its cumulative reward. Most of the prior work on MAB assumes that the reward distributions of each arm are independent. But in a wide variety of decision problems -- from drug dosage to dynamic pricing -- the expected rewards of different arms are correlated, so that selecting one arm provides information about the expected rewards of other arms as well. We propose and analyze a class of models of such decision problems, which we call {\em global bandits}. In the case in which rewards of all arms are deterministic functions of a single unknown parameter, we construct a greedy policy that achieves {\em bounded regret}, with a bound that depends on the single true parameter of the problem. Hence, this policy selects suboptimal arms only finitely many times with probability one. For this case we also obtain a bound on regret that is {\em independent of the true parameter}; this bound is sub-linear, with an exponent that depends on the informativeness of the arms. We also propose a variant of the greedy policy that achieves $\tilde{\mathcal{O}}(\sqrt{T})$ worst-case and $\mathcal{O}(1)$ parameter dependent regret. Finally, we perform experiments on dynamic pricing and show that the proposed algorithms achieve significant gains with respect to the well-known benchmarks.