The ratio of two probability densities, called a density-ratio, is a vital quantity in machine learning.

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.

Each dataset has 513 instances on average. School contains the examination scores of students from 139 schools (datasets). The average outlier rates in a dataset of IoT, Landmine, and School are 0.05, 0.06, and 0.15, For IoT, we randomly chose one target, one validation, and seven source datasets. Algorithm 1 Training procedure of our model for inlier-based outlier detection.Require: LOF and IF use only target unlabeled instances to find outliers. For both methods, the best test AUCs were reported.

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.

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.