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

TL;DR

This paper investigates how the asymmetry of relative entropy affects empirical risk minimization with relative entropy regularization, revealing that it biases solutions towards the reference measure and establishing equivalence between two regularization types.

Contribution

It characterizes the solution to Type-II ERM-RER, compares it with Type-I, and demonstrates the impact of entropy asymmetry on regularization bias and solution support.

Findings

Regularization biases solutions towards the reference measure.

Type-II ERM-RER is equivalent to Type-I with a transformed risk.

Entropy asymmetry influences the support and bias of solutions.

Abstract

The effect of relative entropy asymmetry is analyzed in the context of empirical risk minimization (ERM) with relative entropy regularization (ERM-RER). Two regularizations are considered: $(a)$ the relative entropy of the measure to be optimized with respect to a reference measure (Type-I ERM-RER); and $(b)$ the relative entropy of the reference measure with respect to the measure to be optimized (Type-II ERM-RER). The main result is the characterization of the solution to the Type-II ERM-RER problem and its key properties. By comparing the well-understood Type-I ERM-RER with Type-II ERM-RER, the effects of entropy asymmetry are highlighted. The analysis shows that in both cases, regularization by relative entropy forces the solution's support to collapse into the support of the reference measure, introducing a strong inductive bias that negates the evidence provided by the training data. Finally, it is shown that Type-II regularization is equivalent to Type-I regularization with an appropriate transformation of the empirical risk function.