  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.
