$α$-Divergence Loss Function for Neural Density Ratio Estimation

• URL: http://arxiv.org/abs/2402.02041v3
• Date: Wed, 02 Oct 2024 12:57:41 GMT
• Title: $α$-Divergence Loss Function for Neural Density Ratio Estimation
• Authors: Yoshiaki Kitazawa,
• Abstract summary: Density ratio estimation (DRE) is a fundamental machine learning technique for capturing relationships between two probability distributions. Existing methods face optimization challenges, such as overfitting due to lower-unbounded loss functions, biased mini-batch gradients, vanishing training loss gradients, and high sample requirements for Kullback-Leibler (KL) divergence loss functions. We propose a novel loss function for DRE, the $alpha$-divergence loss function ($alpha$-Div), which is concise but offers stable and effective optimization for DRE.
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• Abstract: Density ratio estimation (DRE) is a fundamental machine learning technique for capturing relationships between two probability distributions. State-of-the-art DRE methods estimate the density ratio using neural networks trained with loss functions derived from variational representations of $f$-divergence. However, existing methods face optimization challenges, such as overfitting due to lower-unbounded loss functions, biased mini-batch gradients, vanishing training loss gradients, and high sample requirements for Kullback-Leibler (KL) divergence loss functions. To address these issues, we focus on $\alpha$-divergence, which provides a suitable variational representation of $f$-divergence. Subsequently, a novel loss function for DRE, the $\alpha$-divergence loss function ($\alpha$-Div), is derived. $\alpha$-Div is concise but offers stable and effective optimization for DRE. The boundedness of $\alpha$-divergence provides the potential for successful DRE with data exhibiting high KL-divergence. Our numerical experiments demonstrate the effectiveness in optimization using $\alpha$-Div. However, the experiments also show that the proposed loss function offers no significant advantage over the KL-divergence loss function in terms of RMSE for DRE. This indicates that the accuracy of DRE is primarily determined by the amount of KL-divergence in the data and is less dependent on $\alpha$-divergence.

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