Empirical Risk Minimization with Generalized Relative Entropy Regularization

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 instead of a probability measure. This assumption leads to a generalization of the ERM-RER (g-ERM-RER) problem that allows for a larger degree of flexibility in the incorporation of prior knowledge over the set of models. The solution of the g-ERM-RER problem is shown to be a unique probability measure mutually absolutely continuous with the reference measure and to exhibit a probably-approximately-correct (PAC) guarantee for the ERM problem. For a given dataset, the empirical risk is shown to be a sub-Gaussian random variable when the models are sampled from the solution to the g-ERM-RER problem. Finally, the sensitivity of the expected empirical risk to deviations from the solution of the g-ERM-RER problem is studied. In particular, the expectation of the absolute value of sensitivity is shown to be upper bounded, up to a constant factor, by the square root of the lautum information between the models and the datasets.