Contextual Multi-Armed Bandits

D´ avid P´ al Abstract We study contextual multi-armed bandit prob- lems where the context comes from a metric space and the payoff satisfies a Lipschitz condi- tion with respect to the metric. Abstractly, a con- textual multi-armed bandit problem models a sit- uation where, in a sequence of independent trials, an online algorithm chooses, based on a given context (side information), an action from a set of possible actions so as to maximize the total pay- off of the chosen actions. The payoff depends on both the action chosen and the context. In con- trast, context-free multi-armed bandit problems, a focus of much previous research, model situa- tions where no side information is available and the payoff depends only on the action chosen. Our problem is motivated by sponsored web search, where the task is to display ads to a user of an Internet search engine based on her search query so as to maximize the click-through rate (CTR) of the ads displayed. We cast this prob- lem as a contextual multi-armed bandit problem where queries and ads form metric spaces and the payoff function is Lipschitz with respect to both the metrics. For any > 0 we present an algorithm with regret O(T a+b+1 a+b+2+ ) where a;b are the covering dimensions of the query space and the ad space respectively. We prove a lower bound ( T ~ a+~+1 ~ a+~+2 ) for the regret of any algo- rithm where ~ ~ are packing dimensions of the query spaces and the ad space respectively. For finite spaces or convex bounded subsets of Eu- clidean spaces, this gives an almost matching up- per and lower bound.