🤖 AI Summary
This work uncovers a critical asymmetry in relative entropy regularization for empirical risk minimization (ERM). We systematically analyze two distinct ERM-relative entropy regularization (ERM-RER) formulations: Type-I, where the optimized measure is regularized relative to a fixed reference measure, and Type-II, where the reference measure is regularized relative to the optimized one. We provide the first structural characterization of Type-II solutions, revealing that they necessarily concentrate—i.e., their support collapses—onto the support of the reference measure, inducing a strong inductive bias. Theoretically, we prove that both types enforce support contraction of the solution, and further establish that Type-II regularization is strictly equivalent to a Type-I formulation under a specific transformation of the loss function. This equivalence refutes the conventional view of Type-I and Type-II as symmetric variants, and advances an information-geometric understanding of how entropy regularization governs generalization.
📝 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.