Ada-BKB: Scalable Gaussian Process Optimization on Continuous Domains by Adaptive Discretization

Marco Rando, Luigi Carratino, Silvia Villa, Lorenzo Rosasco

Abstract: Gaussian process optimization is a successful class of algorithms(e.g. GP-UCB) to optimize a black-box function through sequential evaluations. However, for functions with continuous domains, Gaussian process optimization has to rely on either a fixed discretization of the space, or the solution of a non-convex ptimization subproblem at each evaluation. The first approach can negatively affect performance, while the second approach requires a heavy computational burden. A third option, only recently theoretically studied, is to adaptively discretize the function domain. Even though this approach avoids the extra non-convex optimization costs, the overall computational complexity is still prohibitive. An algorithm such as GP-UCB has a runtime of $\mathcal{O}(T^4)$, where $T$ is the number of iterations. In this paper, we introduce Ada-BKB (Adaptive Budgeted Kernelized Bandit), a no-regret Gaussian process optimization algorithm for functions on continuous domains, that provably runs in $\mathcal{O}(T^2 d_{eff}^2)$, where $d_{eff}$ is the effective dimension of the explored space, and which is typically much smaller than T. We corroborate our theoretical findings with experiments on synthetic non-convex functions and on the real-world problem of hyper-parameter optimization, confirming the good practical performances of the proposed approach.