Gaussian process upper confidence bound (GP-UCB) is a theoretically established algorithm for Bayesian optimization (BO), where we assume the objective function $f$ follows a GP. One notable drawback of GP-UCB is that the theoretical confidence parameter $\beta$ increases along with the iterations and 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 noise 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. Furthermore, we show that randomization plays a key role in avoiding an increase in confidence parameter by showing that GP-UCB using a constant confidence parameter can incur linearly growing expected cumulative regret. 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$\beta$随迭代次数增加而增长，且数值过大。为缓解此缺陷，本文分析了GP-UCB的随机化变体——改进型随机化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，其中期望与概率分别针对$f$及噪声、以及BO算法的随机性进行计算。在两种遗憾分析中，若输入域为有限集，IRGP-UCB可在不增加置信参数的情况下获得次线性遗憾上界。此外，通过证明采用常数置信参数的GP-UCB可能导致线性增长的期望累积遗憾，我们揭示了随机化在避免置信参数增长中的关键作用。最后，我们通过合成函数、基准函数及实际仿真器进行了数值实验验证。