In this paper, the worst-case probability measure over the data is introduced
as a tool for characterizing the generalization capabilities of machine
learning algorithms. More specifically, the worst-case probability measure is a
Gibbs probability measure and the unique solution to the maximization of the
expected loss under a relative entropy constraint with respect to a reference
probability measure. Fundamental generalization metrics, such as the
sensitivity of the expected loss, the sensitivity of the empirical risk, and
the generalization gap are shown to have closed-form expressions involving the
worst-case data-generating probability measure. Existing results for the Gibbs
algorithm, such as characterizing the generalization gap as a sum of mutual
information and lautum information, up to a constant factor, are recovered. A
novel parallel is established between the worst-case data-generating
probability measure and the Gibbs algorithm. Specifically, the Gibbs
probability measure is identified as a fundamental commonality of the model
space and the data space for machine learning algorithms.

该研究介绍了将最坏情况下的概率度量引入数据中，以表征机器学习算法的泛化能力。最坏情况下的概率度量是 Gibbs 概率度量，并且是相对熵约束下期望损失最大化的唯一解。该研究还通过最坏情况下的数据生成概率度量得出了基本的泛化度量指标，例如期望损失的敏感性、经验风险的敏感性和泛化差距，并恢复了 Gibbs 算法的现有结果，将泛化差距的特征化表述为互信息和劳特纳信息之和。而最坏情况下的数据生成概率度量和 Gibbs 算法之间建立了一种新的类比关系，具体而言，Gibbs 概率度量被确定为模型空间和数据空间的机器学习算法的基本共性。