A Dual Optimization View to Empirical Risk Minimisation with f-Divergence Regularization
Une Approche d’Optimisation Duale de la Minimisation du Risque Empirique avec Régularisation par f-Divergence
Résumé
The dual formulation of empirical risk minimization with f-divergence regularization (ERM-f DR) is introduced. The solution of the dual optimization problem to the ERM-f DR is connected to the notion of normalization function introduced as an implicit function. This dual approach leverages the Legendre-Fenchel transform and the implicit function theorem to provide a nonlinear ODE expression to the normalization function. Furthermore, the nonlinear ODE expression and its properties provide a computationally efficient method to calculate the normalization function of the ERM-f DR solution under a mild condition.
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