Efficient Kernelized UCB for Contextual Bandits

In this paper, we tackle the computational efficiency of kernelized UCB algorithms in contextual bandits. While standard methods require a $\mathcal{O}(CT^3)$ complexity where~$T$ is the horizon and the constant $C$ is related to optimizing the UCB rule, we propose an efficient contextual algorithm for large-scale problems. Specifically, our method relies on incremental Nystr\"om approximations of the joint kernel embedding of contexts and actions. This allows us to achieve a complexity of $\mathcal{O}(CTm^2)$ where $m$ is the number of Nystr\"om points. To recover the same regret as the standard kernelized UCB algorithm, $m$ needs to be of order of the effective dimension of the problem, which is at most $\mathcal{O}(\sqrt{T})$ and nearly constant in some cases.