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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 optimizationhas to rely on either a fixed discretization of the space, or the solution of anon-convex optimization subproblem at each evaluation. The first approach cannegatively affect performance, while the second approach requires a heavycomputational burden. A third option, only recently theoretically studied, isto adaptively discretize the function domain. Even though this approach avoidsthe extra non-convex optimization costs, the overall computational complexityis still prohibitive. An algorithm such as GP-UCB has a runtime of $O(T^4)$,where $T$ is the number of iterations. In this paper, we introduce Ada-BKB(Adaptive Budgeted Kernelized Bandit), a no-regret Gaussian processoptimization algorithm for functions on continuous domains, that provably runsin $O(T^2 d_\text{eff}^2)$, where $d_\text{eff}$ is the effective dimension ofthe explored space, and which is typically much smaller than $T$. Wecorroborate our theoretical findings with experiments on synthetic non-convexfunctions and on the real-world problem of hyper-parameter optimization,confirming the good practical performances of the proposed approach.

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