Contextual Bandits with Cross-Learning

Abstract

In the classic contextual bandits problem, in each round t, a learner observes some context c, chooses some action i to perform, and receives some reward [Formula: see text]. We consider the variant of this problem in which in addition to receiving the reward [Formula: see text], the learner also learns the values of [Formula: see text] for some other contexts [Formula: see text] in set [Formula: see text], that is, the rewards that would be achieved by performing that action under different contexts [Formula: see text]. This variant arises in several strategic settings, such as learning how to bid in nontruthful repeated auctions, which has gained a lot of attention lately as many platforms have switched to running first price auctions. We call this problem the contextual bandits problem with cross-learning. The best algorithms for the classic contextual bandits problem achieve [Formula: see text] regret against all stationary policies, in which C is the number of contexts, K the number of actions, and T the number of rounds. We design and analyze new algorithms for the contextual bandits problem with cross-learning and show that their regret has better dependence on the number of contexts. Under complete cross-learning in which the rewards for all contexts are learned when choosing an action, that is, set [Formula: see text] contains all contexts, we show that our algorithms achieve regret [Formula: see text], removing the dependence on C. For any other cases, that is, under partial cross-learning in which [Formula: see text] for some context–action pair of (i, c), the regret bounds depend on how the sets [Formula: see text] impact the degree to which cross-learning between contexts is possible. We simulate our algorithms on real auction data from an ad exchange running first price auctions and show that they outperform traditional contextual bandit algorithms.