  Functions of the ratio of the densities $p/q$ are widely used in machine
learning to quantify the discrepancy between the two distributions $p$ and $q$.
For high-dimensional distributions, binary classification-based density ratio
estimators have shown great promise. However, when densities are well
separated, estimating the density ratio with a binary classifier is
challenging. In this work, we show that the state-of-the-art density ratio
estimators perform poorly on well-separated cases and demonstrate that this is
due to distribution shifts between training and evaluation time. We present an
alternative method that leverages multi-class classification for density ratio
estimation and does not suffer from distribution shift issues. The method uses
a set of auxiliary densities $\{m_k\}_{k=1}^K$ and trains a multi-class
logistic regression to classify the samples from $p, q$, and $\{m_k\}_{k=1}^K$
into $K+2$ classes. We show that if these auxiliary densities are constructed
such that they overlap with $p$ and $q$, then a multi-class logistic regression
allows for estimating $\log p/q$ on the domain of any of the $K+2$
distributions and resolves the distribution shift problems of the current
state-of-the-art methods. We compare our method to state-of-the-art density
ratio estimators on both synthetic and real datasets and demonstrate its
superior performance on the tasks of density ratio estimation, mutual
information estimation, and representation learning. Code:
https://www.blackswhan.com/mdre/
