## Abstract

The empirical risk minimization (ERM) problem with relative entropy regularization (ERM-RER) is investigated under the assumption that the reference measure is a σ-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.

Original language | English |
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Pages (from-to) | 1 |

Number of pages | 1 |

Journal | IEEE Transactions on Information Theory |

DOIs | |

State | Accepted/In press - 2024 |

## Keywords

- Empirical Risk Minimization
- Entropy
- Generalization
- Gibbs Algorithm
- Gibbs Measure
- Measurement uncertainty
- PAC-Learning
- Probability distribution
- Q measurement
- Random variables
- Relative Entropy Regularization
- Risk minimization
- Sensitivity
- Sensitivity
- Supervised Learning