Analysis of the Relative Entropy Asymmetry in the Regularization of Empirical Risk Minimization

Francisco Daunas; Iñaki Esnaola; Samir M. Perlaza; H. Vincent Poor

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arXiv · 2306.07123

Analysis of the Relative Entropy Asymmetry in the Regularization of Empirical Risk Minimization

Francisco Daunas·Iñaki Esnaola·Samir M. Perlaza·H. Vincent Poor

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Abstract

The effect of the relative entropy asymmetry is analyzed in the empirical risk minimization with relative entropy regularization (ERM-RER) problem. A novel regularization is introduced, coined Type-II regularization, that allows for solutions to the ERM-RER problem with a support that extends outside the support of the reference measure. The solution to the new ERM-RER Type-II problem is analytically characterized in terms of the Radon-Nikodym derivative of the reference measure with respect to the solution. The analysis of the solution unveils the following properties of relative entropy when it acts as a regularizer in the ERM-RER problem: i) relative entropy forces the support of the Type-II solution to collapse into the support of the reference measure, which introduces a strong inductive bias that dominates the evidence provided by the training data; ii) Type-II regularization is equivalent to classical relative entropy regularization with an appropriate transformation of the empirical risk function. Closed-form expressions of the expected empirical risk as a function of the regularization parameters are provided.

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Francisco Daunas, Iñaki Esnaola, Samir M. Perlaza, H. Vincent Poor. 2023-06-12. Analysis of the Relative Entropy Asymmetry in the Regularization of Empirical Risk Minimization. https://arxiv.org/abs/2306.07123

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