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Gaussian processes (GP) are a well studied Bayesian approach for theoptimization of black-box functions. Despite their effectiveness in simpleproblems, GP-based algorithms hardly scale to high-dimensional functions, astheir per-iteration time and space cost is at least quadratic in the number ofdimensions $d$ and iterations $t$. Given a set of $A$ alternatives to choosefrom, the overall runtime $O(t^3A)$ is prohibitive. In this paper we introduceBKB (budgeted kernelized bandit), a new approximate GP algorithm foroptimization under bandit feedback that achieves near-optimal regret (and hencenear-optimal convergence rate) with near-constant per-iteration complexity andremarkably no assumption on the input space or covariance of the GP.  We combine a kernelized linear bandit algorithm (GP-UCB) with randomizedmatrix sketching based on leverage score sampling, and we prove that randomlysampling inducing points based on their posterior variance gives an accuratelow-rank approximation of the GP, preserving variance estimates and confidenceintervals. As a consequence, BKB does not suffer from variance starvation, animportant problem faced by many previous sparse GP approximations. Moreover, weshow that our procedure selects at most $\tilde{O}(d_{eff})$ points, where$d_{eff}$ is the effective dimension of the explored space, which is typicallymuch smaller than both $d$ and $t$. This greatly reduces the dimensionality ofthe problem, thus leading to a $O(TAd_{eff}^2)$ runtime and $O(A d_{eff})$space complexity.

Gaussian Process Optimization with Adaptive Sketching: Scalable and No Regret