A Dual Optimization View to Empirical Risk Minimization with f-Divergence Regularization
arXiv:2508.03314v1 Announce Type: new
Abstract: The dual formulation of empirical risk minimization with f-divergence regularization (ERM-fDR) is introduced. The solution of the dual optimization problem to the ERM-fDR 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-fDR solution under a mild condition.
Abstract: The dual formulation of empirical risk minimization with f-divergence regularization (ERM-fDR) is introduced. The solution of the dual optimization problem to the ERM-fDR 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-fDR solution under a mild condition.
Francisco Daunas, I~naki Esnaola, Samir M. Perlaza
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