Gaussian process upper confidence bound (GP-UCB) is a theoretically established algorithm for Bayesian optimization (BO), where we assume the objective function $f$ follows GP. One notable drawback of GP-UCB is that the theoretical confidence parameter $\beta$ increased along with the iterations is too large. To alleviate this drawback, this paper analyzes the randomized variant of GP-UCB called improved randomized GP-UCB (IRGP-UCB), which uses the confidence parameter generated from the shifted exponential distribution. We analyze the expected regret and conditional expected regret, where the expectation and the probability are taken respectively with $f$ and noises and with the randomness of the BO algorithm. In both regret analyses, IRGP-UCB achieves a sub-linear regret upper bound without increasing the confidence parameter if the input domain is finite. Finally, we show numerical experiments using synthetic and benchmark functions and real-world emulators.

翻译：高斯过程上置信界（GP-UCB）是贝叶斯优化（BO）中一种理论完备的算法，其假设目标函数$f$服从高斯过程。GP-UCB的一个显著缺陷是其理论置信参数$\beta$随迭代次数增加而变得过大。为缓解此缺陷，本文分析了GP-UCB的随机化变体——改进随机化GP-UCB（IRGP-UCB），该算法采用移位指数分布生成的置信参数。我们分析了期望遗憾与条件期望遗憾，其中期望和概率分别对$f$与噪声、以及对BO算法的随机性进行计算。在两种遗憾分析中，当输入域有限时，IRGP-UCB在不增加置信参数的情况下实现了次线性遗憾上界。最后，我们通过合成函数、基准函数和实际仿真器进行了数值实验验证。