  Gaussian process upper confidence bound (GP-UCB) is a theoretically promising
approach for black-box optimization; however, the confidence parameter $\beta$
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
$\beta$. 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.
