Generalization Analysis of Machine Learning Algorithms via the Worst-Case Data-Generating Probability Measure
Xinying Zou, Samir M. Perlaza, Iñaki Esnaola, Eitan Altman
Abstract
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.
BibTeX
@article{Zou_Perlaza_Esnaola_Altman_2024, title={Generalization Analysis of Machine Learning Algorithms via the Worst-Case Data-Generating Probability Measure}, volume={38}, url={https://ojs.aaai.org/index.php/AAAI/article/view/29674}, DOI={10.1609/aaai.v38i15.29674}, abstractNote={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.}, number={15}, journal={Proceedings of the AAAI Conference on Artificial Intelligence}, author={Zou, Xinying and Perlaza, Samir M. and Esnaola, Iñaki and Altman, Eitan}, year={2024}, month={Mar.}, pages={17271-17279} }