Bounds on $L_p$ Errors in Density Ratio Estimation via $f$-Divergence Loss Functions

• URL: http://arxiv.org/abs/2410.01516v1
• Date: Wed, 2 Oct 2024 13:05:09 GMT
• Title: Bounds on $L_p$ Errors in Density Ratio Estimation via $f$-Divergence Loss Functions
• Authors: Yoshiaki Kitazawa,
• Abstract summary: Density ratio estimation (DRE) is a fundamental machine learning technique for identifying relationships between two probability distributions. $f$-divergence loss functions, derived from variational representations of $f$-divergence, are commonly employed in DRE to achieve state-of-the-art results. This study presents a novel perspective on DRE using $f$-divergence loss functions by deriving the upper and lower bounds on $L_p$ errors.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Density ratio estimation (DRE) is a fundamental machine learning technique for identifying relationships between two probability distributions. $f$-divergence loss functions, derived from variational representations of $f$-divergence, are commonly employed in DRE to achieve state-of-the-art results. This study presents a novel perspective on DRE using $f$-divergence loss functions by deriving the upper and lower bounds on $L_p$ errors. These bounds apply to any estimator within a class of Lipschitz continuous estimators, irrespective of the specific $f$-divergence loss functions utilized. The bounds are formulated as a product of terms that include the data dimension and the expected value of the density ratio raised to the power of $p$. Notably, the lower bound incorporates an exponential term dependent on the Kullback--Leibler divergence, indicating that the $L_p$ error significantly increases with the Kullback--Leibler divergence for $p > 1$, and this increase becomes more pronounced as $p$ increases. Furthermore, these theoretical findings are substantiated through numerical experiments.

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