  The ratio of two probability densities, called a density-ratio, is a vital
quantity in machine learning. In particular, a relative density-ratio, which is
a bounded extension of the density-ratio, has received much attention due to
its stability and has been used in various applications such as outlier
detection and dataset comparison. Existing methods for (relative) density-ratio
estimation (DRE) require many instances from both densities. However,
sufficient instances are often unavailable in practice. In this paper, we
propose a meta-learning method for relative DRE, which estimates the relative
density-ratio from a few instances by using knowledge in related datasets.
Specifically, given two datasets that consist of a few instances, our model
extracts the datasets' information by using neural networks and uses it to
obtain instance embeddings appropriate for the relative DRE. We model the
relative density-ratio by a linear model on the embedded space, whose global
optimum solution can be obtained as a closed-form solution. The closed-form
solution enables fast and effective adaptation to a few instances, and its
differentiability enables us to train our model such that the expected test
error for relative DRE can be explicitly minimized after adapting to a few
instances. We empirically demonstrate the effectiveness of the proposed method
by using three problems: relative DRE, dataset comparison, and outlier
detection.
