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.