Randomized Gaussian Process Upper Confidence Bound with Tighter Bayesian Regret Bounds
This work addresses a theoretical bottleneck in Bayesian optimization for researchers and practitioners, offering incremental improvements to regret bounds.
The paper tackles the issue of large confidence parameters in Gaussian process upper confidence bound (GP-UCB) methods for black-box optimization by proposing an improved randomized version (IRGP-UCB) based on a two-parameter exponential distribution, achieving tighter Bayesian regret bounds and avoiding over-exploration in later iterations.
Gaussian process upper confidence bound (GP-UCB) is a theoretically promising approach for black-box optimization; however, the confidence parameter $β$ is considerably large in the theorem and chosen heuristically in practice. Then, randomized GP-UCB (RGP-UCB) uses a randomized confidence parameter, which follows the Gamma distribution, to mitigate the impact of manually specifying $β$. This study first generalizes the regret analysis of RGP-UCB to a wider class of distributions, including the Gamma distribution. Furthermore, we propose improved RGP-UCB (IRGP-UCB) based on a two-parameter exponential distribution, which achieves tighter Bayesian regret bounds. IRGP-UCB does not require an increase in the confidence parameter in terms of the number of iterations, which avoids over-exploration in the later iterations. Finally, we demonstrate the effectiveness of IRGP-UCB through extensive experiments.