  Among various acquisition functions (AFs) in Bayesian optimization (BO),
Gaussian process upper confidence bound (GP-UCB) and Thompson sampling (TS) are
well-known options with established theoretical properties regarding Bayesian
cumulative regret (BCR). Recently, it has been shown that a randomized variant
of GP-UCB achieves a tighter BCR bound compared with GP-UCB, which we call the
tighter BCR bound for brevity. Inspired by this study, this paper first shows
that TS achieves the tighter BCR bound. On the other hand, GP-UCB and TS often
practically suffer from manual hyperparameter tuning and over-exploration
issues, respectively. Therefore, we analyze yet another AF called a probability
of improvement from the maximum of a sample path (PIMS). We show that PIMS
achieves the tighter BCR bound and avoids the hyperparameter tuning, unlike
GP-UCB. Furthermore, we demonstrate a wide range of experiments, focusing on
the effectiveness of PIMS that mitigates the practical issues of GP-UCB and TS.
