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

• URL: http://arxiv.org/abs/2305.00869v1
• Date: Mon, 1 May 2023 15:10:56 GMT
• Title: Estimating the Density Ratio between Distributions with High Discrepancy using Multinomial Logistic Regression
• Authors: Akash Srivastava, Seungwook Han, Kai Xu, Benjamin Rhodes, Michael U. Gutmann
• Abstract summary: We show that the state-of-the-art density ratio estimators perform poorly on well-separated cases. We present an alternative method that leverages multi-class classification for density ratio estimation.
• Score: 21.758330613138778
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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/

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