Gaussian Process Optimization with Adaptive Sketching: Scalable and No Regret

Abstract

Gaussian processes (GP) are a stochastic processes, used as 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 \emph{quadratic} in the number of dimensions $d$ and iterations $t$. Given a set of $A$ alternatives to choose from, the overall runtime $O(t^3 A)$ is prohibitive. In this paper, we introduce BKB (\textit{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) leverage score sampling as a randomized matrix sketching and prove that selecting 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 \emph{variance starvation}, an important problem faced by many previous sparse GP approximations. Moreover, we show that our procedure selects at most $\widetilde{O}(d_{eff})$ points, where $d_{eff}$ is the \emph{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 $O(T A d_{eff}^2)$ runtime and $O(A d_{eff})$ space complexity.