Bounds on Lp errors in density ratio estimation via f-divergence loss functions

Density ratio estimation (DRE) is a core technique in machine learning used to capture relationships between two probability distributions. f-divergence loss functions, which are derived from variational representations of f-divergence, have become a standard choice in DRE for achieving cutting-edge performance. This study provides novel theoretical insights into DRE by deriving upper and lower bounds on the Lp errors through f-divergence loss functions. These bounds apply to any estimator belonging to a class of Lipschitz continuous estimators, irrespective of the specific f-divergence loss function employed. The derived bounds are expressed as a product involving the data dimensionality and the expected value of the density ratio raised to the p-th power. Notably, the lower bound includes an exponential term that depends on the Kullback--Leibler (KL) divergence, revealing that the Lp error increases significantly as the KL divergence grows when p > 1. This increase becomes even more pronounced as the value of p grows. The theoretical insights are validated through numerical experiments.