Gaussian processes (GPs) are the most common formalism for defining
probability distributions over spaces of functions. While applications of GPs
are myriad, a comprehensive understanding of GP sample paths, i.e. the function
spaces over which they define a probability measure on, is lacking. In
practice, GPs are not constructed through a probability measure, but instead
through a mean function and a covariance kernel. In this paper we provide
necessary and sufficient conditions on the covariance kernel for the sample
paths of the corresponding GP to attain a given regularity. We use the
framework of H\"older regularity as it grants us particularly straightforward
conditions, which simplify further in the cases of stationary and isotropic
GPs. We then demonstrate that our results allow for novel and unusually tight
characterisations of the sample path regularities of the GPs commonly used in
machine learning applications, such as the Mat\'ern GPs.

通过对协方差核函数进行充分的必要条件判断，我们揭示了高斯过程的样本路径达到特定规则的条件，并证明了这些结果对机器学习中常用的 Matern 高斯过程等样本路径规则的特征提供了新颖而紧凑的描述。

高斯过程的协方差核采样路径规律

Sample Path Regularity of Gaussian Processes from the Covariance Kernel

This works extends the Random Embedding Bayesian Optimization approach by
integrating a warping of the high dimensional subspace within the covariance
kernel. The proposed warping, that relies on elementary geometric
considerations, allows mitigating the drawbacks of the high extrinsic
dimensionality while avoiding the algorithm to evaluate points giving redundant
information. It also alleviates constraints on bound selection for the embedded
domain, thus improving the robustness, as illustrated with a test case with 25
variables and intrinsic dimension 6.

该论文提出了一种基于随机嵌入贝叶斯优化的方法，并通过将高维子空间的扭曲集成到协方差核中来扩展它。该方法可以解决高外部维度的缺点，避免算法评估提供冗余信息的点，并缓解嵌入域选择边界限制，从而提高其鲁棒性，并在一个具有 25 个变量和内部维度 6 的测试用例中进行了说明。

通过随机嵌入改进贝叶斯优化的扭曲核提高鲁棒性

A warped kernel improving robustness in Bayesian optimization via random  embeddings

In this paper we develop a dynamic form of Bayesian optimization for machine
learning models with the goal of rapidly finding good hyperparameter settings.
Our method uses the partial information gained during the training of a machine
learning model in order to decide whether to pause training and start a new
model, or resume the training of a previously-considered model. We specifically
tailor our method to machine learning problems by developing a novel
positive-definite covariance kernel to capture a variety of training curves.
Furthermore, we develop a Gaussian process prior that scales gracefully with
additional temporal observations. Finally, we provide an information-theoretic
framework to automate the decision process. Experiments on several common
machine learning models show that our approach is extremely effective in
practice.

本文介绍一种基于贝叶斯优化的机器学习超参数搜索方法，在训练过程中根据部分信息决定是否暂停、新建模型，以及恢复之前考虑过的模型。该方法特别适合机器学习问题，通过开发一种新的正定协方差核来捕捉多种训练曲线，并开发了一个能够与额外时间观察优雅扩展的高斯过程先验。此外，我们提供了一个信息理论的框架来自动化决策过程。在多种常见的机器学习模型上的实验表明，我们的方法在实践中非常有效。