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

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: (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</i>) the relative entropy of the measure to be optimized with respect to a reference measure (Type-I ERM-RER); and (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i>) 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 support of the solution 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.