Estimating the Density Ratio between Distributions with High Discrepancy using Multinomial Logistic Regression

Akash Srivastava, Seungwook Han, Kai Xu, Benjamin Rhodes, Michael U. Gutmann

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

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 {mk} Kk=1 and trains a multi-class logistic regression to classify the samples from p, q and {mk} Kk=1 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/
Original languageEnglish
Pages (from-to)1-23
JournalTransactions on Machine Learning Research
Volume2023
Issue number3
Publication statusPublished - 4 Apr 2023

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