A Domain-Shrinking based Bayesian Optimization Algorithm with Order-Optimal Regret Performance
This provides a more efficient Bayesian optimization method for machine learning practitioners dealing with high-dimensional function optimization, though it is incremental as it builds on existing GP-based approaches.
The paper tackles sequential optimization of unknown functions in reproducing kernel Hilbert spaces by proposing a Gaussian process-based algorithm with domain shrinking through tree-based pruning, achieving order-optimal regret performance up to a poly-logarithmic factor and reducing computational complexity by a factor of O(T^{2d-1}) compared to GP-UCB algorithms.
We consider sequential optimization of an unknown function in a reproducing kernel Hilbert space. We propose a Gaussian process-based algorithm and establish its order-optimal regret performance (up to a poly-logarithmic factor). This is the first GP-based algorithm with an order-optimal regret guarantee. The proposed algorithm is rooted in the methodology of domain shrinking realized through a sequence of tree-based region pruning and refining to concentrate queries in increasingly smaller high-performing regions of the function domain. The search for high-performing regions is localized and guided by an iterative estimation of the optimal function value to ensure both learning efficiency and computational efficiency. Compared with the prevailing GP-UCB family of algorithms, the proposed algorithm reduces computational complexity by a factor of $O(T^{2d-1})$ (where $T$ is the time horizon and $d$ the dimension of the function domain).