The empirical risk minimization (ERM) problem with relative entropy regularization (ERM-RER) is investigated under the assumption that the reference measure is a $\sigma$-finite measure, and not necessarily a probability measure. Under this assumption, which leads to a generalization of the ERM-RER problem allowing a larger degree of flexibility for incorporating prior knowledge, numerous relevant properties are stated. Among these properties, the solution to this problem, if it exists, is shown to be a unique probability measure, mutually absolutely continuous with the reference measure. Such a solution exhibits a probably-approximately-correct guarantee for the ERM problem independently of whether the latter possesses a solution. For a fixed dataset and under a specific condition, the empirical risk is shown to be a sub-Gaussian random variable when the models are sampled from the solution to the ERM-RER problem. The generalization capabilities of the solution to the ERM-RER problem (the Gibbs algorithm) are studied via the sensitivity of the expected empirical risk to deviations from such a solution towards alternative probability measures. Finally, an interesting connection between sensitivity, generalization error, and lautum information is established.

翻译：研究了在参考测度为σ有限测度（不一定是概率测度）的假设下，带相对熵正则化的经验风险最小化（ERM-RER）问题。在此假设下，ERM-RER问题得到泛化，从而允许在融入先验知识时具有更大灵活性，并阐述了若干相关性质。在这些性质中，若解存在，则其被证明是唯一的概率测度，且与参考测度相互绝对连续。该解为经验风险最小化问题提供了一种概率近似正确（PAC）保证，无论原问题是否存在解。对于固定数据集，在特定条件下，当模型从ERM-RER问题的解中采样时，经验风险被证明是次高斯随机变量。通过研究期望经验风险对偏离该解而趋向其他概率测度的敏感性，分析了ERM-RER问题解（吉布斯算法）的泛化能力。最后，建立了敏感性、泛化误差与Lautum信息之间有趣的联系。