Non-Negative Bregman Divergence Minimization for Deep Direct Density Ratio Estimation

The estimation of the ratio of two probability densities has garnered attention as the density ratio is useful in various machine learning tasks, such as anomaly detection and domain adaptation. To estimate the density ratio, methods collectively known as direct density ratio estimation (DRE) have been explored. These methods are based on the minimization of the Bregman (BR) divergence between a density ratio model and the true density ratio. However, existing direct DRE suffers from serious overfitting when using flexible models such as neural networks. In this paper, we introduce a non-negative correction for empirical risk using only the prior knowledge of the upper bound of the density ratio. This correction makes a DRE method more robust against overfitting and enables the use of flexible models. In the theoretical analysis, we discuss the consistency of the empirical risk. In our experiments, the proposed estimators show favorable performance in inlier-based outlier detection and covariate shift adaptation.