Approximation Methods for Kernelized Bandits

The RKHS bandit problem (also called kernelized multi-armed bandit problem) is an online optimization problem of non-linear functions with noisy feedbacks. Most of the existing methods for the problem have sub-linear regret guarantee at the cost of high computational complexity. For example, IGP-UCB requires at least quadratic time in the number of observed samples at each round. In this paper, using deep results provided by the approximation theory, we approximately reduce the problem to the well-studied linear bandit problem of an appropriate dimension. Then, we propose several algorithms and prove that they achieve comparable regret guarantee to the existing methods (GP-UCB, IGP-UCB) with less computational complexity. Specifically, our proposed methods require polylogarithmic time to select an arm at each round for kernels with "infinite smoothness" (e.g. the rational quadratic and squared exponential kernels). Furthermore, we empirically show our proposed method has comparable regret to the existing method and its running time is much shorter.