Bayesian Optimization (BO) is widely used for optimising black-box functions but requires us to specify the length scale hyperparameter, which defines the smoothness of the functions the optimizer will consider. Most current BO algorithms choose this hyperparameter by maximizing the marginal likelihood of the observed data, albeit risking misspecification if the objective function is less smooth in regions we have not yet explored. The only prior solution addressing this problem with theoretical guarantees was A-GP-UCB, proposed by Berkenkamp et al. (2019). This algorithm progressively decreases the length scale, expanding the class of functions considered by the optimizer. However, A-GP-UCB lacks a stopping mechanism, leading to over-exploration and slow convergence. To overcome this, we introduce Length scale Balancing (LB) - a novel approach, aggregating multiple base surrogate models with varying length scales. LB intermittently adds smaller length scale candidate values while retaining longer scales, balancing exploration and exploitation. We formally derive a cumulative regret bound of LB and compare it with the regret of an oracle BO algorithm using the optimal length scale. Denoting the factor by which the regret bound of A-GP-UCB was away from oracle as $g(T)$, we show that LB is only $\log g(T)$ away from oracle regret. We also empirically evaluate our algorithm on synthetic and real-world benchmarks and show it outperforms A-GP-UCB, maximum likelihood estimation and MCMC.

翻译：贝叶斯优化（BO）被广泛用于优化黑箱函数，但需要指定长度尺度超参数，该参数定义了优化器所考虑函数的平滑度。当前大多数BO算法通过最大化观测数据的边际似然来选择该超参数，然而若目标函数在未探索区域平滑性较低，则存在设定错误的风险。此前唯一具有理论保证的解决方案是Berkenkamp等人（2019）提出的A-GP-UCB算法。该算法逐步减小长度尺度，从而扩展优化器所考虑的函数类别。然而，A-GP-UCB缺乏停止机制，导致过度探索和收敛缓慢。为克服这一问题，我们提出了长度尺度平衡（LB）——一种通过聚合具有不同长度尺度的多个基础代理模型的新方法。LB在保留较长尺度的同时间歇性地添加较小长度尺度的候选值，从而平衡探索与利用。我们形式化推导了LB的累积遗憾界，并将其与使用最优长度尺度的预言机BO算法的遗憾进行比较。令A-GP-UCB的遗憾界偏离预言机的因子为$g(T)$，我们证明LB仅偏离预言机遗憾$\log g(T)$。我们还在合成与真实世界基准测试中对算法进行了实证评估，结果表明其性能优于A-GP-UCB、最大似然估计和MCMC方法。