Gaussian processes (GP) are a popular Bayesian approach for the optimization of black-box functions. Despite their effectiveness in simple problems, GP-based algorithms hardly scale to complex 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 $A$ alternative to choose from, the overall runtime $O(t^3A)$ quickly becomes prohibitive. In this paper, we introduce BKB (budgeted kernelized bandit), an 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 no assumption on the input space or the GP’s covariance. Combining a kernelized linear bandit algorithm (GP-UCB) with randomized matrix sketching technique (i.e., leverage score sampling), we 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 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 $\widetilde{O}(TAd_{eff}^2)$ runtime and $\widetilde{O}(A d_{eff})$ space complexity.