Linear bandits in high dimension and recommendation systems

A large number of online services provide automated recommendations to help users to navigate through a large collection of items. New items (products, videos, songs, advertisements) are suggested on the basis of the user's past history and - when available - her demographic profile. Recommendations have to satisfy the dual goal of helping the user to explore the space of available items, while allowing the system to probe the user's preferences. We model this trade-off using linearly parametrized multi-armed bandits and prove upper and lower bounds that coincide up to constants in the data poor (high-dimensional) regime. We test (a variation of) the scheme used for estabilishing achievability on the Netflix dataset, and obtain results in agreement with the theory.