Contextual Bandits with Cross-learning

In the classical contextual bandits problem, in each round t, a learner observes some context c, chooses some action i to perform, and receives some reward ri,t(c). We consider the variant of this problem where in addition to receiving the reward ri,t(c), the learner also learns the values of ri,t(c') for some other contexts c' in set Oi(c); i.e., the rewards that would have been achieved by performing that action under different contexts c'∈ Oi(c). This variant arises in several strategic settings, such as learning how to bid in non-truthful 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 classical contextual bandits problem achieve tildeO(√CKT) regret against all stationary policies, where 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 where the rewards for all contexts are learned when choosing an action, i.e., set Oi(c) contains all contexts, we show that our algorithms achieve regret tildeO(√KT), removing the dependence on C. For any other cases, i.e., under partial cross-learning where |Oi(c)|< C for some context-action pair of (i,c), the regret bounds depend on how the sets mathcal Oi(c) 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.