Gaussian process optimization is a successful class of algorithms (e.g. GP-UCB) to optimize a black-box function through sequential evaluations. However, when the domain of the function is continuous, Gaussian process optimization has to either rely on a fixed discretization of the space, or solve a non-convex optimization subproblem at each evaluation. The first approach can negatively affect performance, while the second one puts a heavy computational burden on the algorithm. A third option, that only recently has been 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 $O(T^4)$, 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 $O(T^2 d_\text{eff}^2)$, where $d_\text{eff}$ is the effective dimension of the explored space, and which is typically much smaller than $T$. We corroborate our findings with experiments on synthetic non-convex functions and on the real-world problem of hyper-parameter optimization.

翻译：Gausian 进程优化是一个成功的算法类别(如 GP- UCB ), 以便通过连续评估优化黑盒功能。 但是, 当函数的域是连续的时, Gaussian 进程优化必须依靠空间的固定离散化, 或者在每次评估中解决非conx优化子问题。 第一种方法可能会对性能产生消极影响, 而第二种方法会给算法带来沉重的计算负担。 第三个方案, 只是在最近才在理论上研究过的, 是适应性地将功能域分解。 即使这种方法避免了额外的非conx优化成本, 总体计算复杂性仍然令人望而却望而却步。 GP- UCB 等算法的运行时间是$O (T+4), 或解决非conx 优化的运行时间是 $(T+2+xx) 的运行时间, 美元是重复数。 在此文件中, 我们介绍Ada- BKB (Adaptab) (Admatrial 预算型的Kennelizelization Band), 一个无报复的功能优化算算算法, 在 $Oe- $Ox slupluplus prilal_ dx brouplus