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

Recently, neural networks have produced state-of-the-art results for density-ratio estimation (DRE), a fundamental technique in machine learning. However, existing methods bear optimization issues that arise from the loss functions of DRE: a large sample requirement of Kullback--Leibler (KL)-divergence, vanishing of train loss gradients, and biased gradients of the loss functions. Thus, an $\alpha$-divergence loss function ($\alpha$-Div) that offers concise implementation and stable optimization is proposed in this paper. Furthermore, technical justifications for the proposed loss function are presented. The stability of the proposed loss function is empirically demonstrated and the estimation accuracy of DRE tasks is investigated. Additionally, this study presents a sample requirement for DRE using the proposed loss function in terms of the upper bound of $L_1$ error, which connects a curse of dimensionality as a common problem in high-dimensional DRE tasks.