Efficient Linear Bandits through Matrix Sketching

We prove that two popular linear contextual bandit algorithms, OFUL and Thompson Sampling, can be made efficient using Frequent Directions, a deterministic online sketching technique. More precisely, we show that a sketch of size m allows a O(md) update time for both algorithms, as opposed to Ω(d 2 ) required by their non-sketched versions in general (where d is the dimension of context vectors). This computational speedup is accompanied by regret bounds of order (1 + εm) 3/2d √ T for OFUL and of order (1 + εm)d 3/2√ T for Thompson Sampling, where εm is bounded by the sum of the tail eigenvalues not covered by the sketch. In particular, when the selected contexts span a subspace of dimension at most m, our algorithms have a regret bound matching that of their slower, non-sketched counterparts. Experiments on real-world datasets corroborate our theoretical results.