Gaussian processes (GP) are a well studied Bayesian approach for the
optimization of black-box functions. Despite their effectiveness in simple
problems, GP-based algorithms hardly scale to high-dimensional functions, as
their per-iteration time and space cost is at least quadratic in the number of
dimensions d and iterations t. Given a set of A alternatives to choose
from, the overall runtime O(t3A) is prohibitive. In this paper we introduce
BKB (budgeted kernelized bandit), a new approximate GP algorithm for
optimization under bandit feedback that achieves near-optimal regret (and hence
near-optimal convergence rate) with near-constant per-iteration complexity and
remarkably no assumption on the input space or covariance of the GP.
We combine a kernelized linear bandit algorithm (GP-UCB) with randomized
matrix sketching based on leverage score sampling, and we prove that randomly
sampling inducing points based on their posterior variance gives an accurate
low-rank approximation of the GP, preserving variance estimates and confidence
intervals. As a consequence, BKB does not suffer from variance starvation, an
important problem faced by many previous sparse GP approximations. Moreover, we
show that our procedure selects at most O~(deff) points, where
deff is the effective dimension of the explored space, which is typically
much smaller than both d and t. This greatly reduces the dimensionality of
the problem, thus leading to a O(TAdeff2) runtime and O(Adeff)
space complexity.Comment: Accepted at COLT 2019. Corrected typos and improved comparison with
existing method