改进的随机高斯过程算法避免了置信参数增长，实现更优优化性能。
Regret Analysis for Randomized Gaussian Process Upper Confidence Bound
- 用移位指数分布生成置信参数，实现随机化策略。
- 在有限输入域下，累积遗憾呈次线性增长，无需增大置信参数。
- 适合关注理论保证与稳定贝叶斯优化的科研人员。
高斯过程上置信界（GP-UCB）是贝叶斯优化中一个理论严谨的算法，假设目标函数 $f$ 服从高斯过程。其显著缺点是理论置信参数 $β$ 随迭代次数增加而增大，导致过大。本文分析了改进的随机化变体——改进随机化 GP-UCB（IRGP-UCB），该方法采用从移位指数分布生成的置信参数。我们分析了期望遗憾和条件期望遗憾，分别对函数 $f$、噪声以及优化算法的随机性取期望。在两种分析中，若输入域有限，IRGP-UCB 均可实现不随迭代增长的置信参数下的次线性遗憾上界。此外，我们证明随机化在避免置信参数上升中起关键作用：使用固定置信参数的普通 GP-UCB 会引发线性增长的期望累积遗憾。最后，通过合成函数、基准函数及真实世界模拟器的数值实验验证了该方法的有效性。
原文摘要 · Abstract (English)
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 $β$ 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.
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