Bounds on Lp Errors in Density Ratio Estimation via f-Divergence Loss Functions
Yoshiaki Kitazawa
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 $L_p$ 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 $L_p$ 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.
density ratio estimation, variational divergence optimization, Kullback–Leibler divergence, $f$-divergence, $L_p$ error, the curse of dimensionality, GAN
We provide both lower  and upper  bounds on $L_p$ errors in DRE, which hold for any member of a group of Lipschitz continuous estimators regardless of the specific $f$-divergence loss functions used.