Learning from Totally Unpaired Data via Minimum Conditional Entropy Coupling

Abstract

Consider an "unsupervised classification" setting, where the goal is to learn a predictor of the categorical class variable $X$ using the categorical feature variables $Y_1,\dots ,Y_m$, but we can only train the predictor on $m+1$ totally unpaired datasets, where each dataset only consists of unordered samples of one of the variables $X,Y_1,\dots ,Y_m$. Although this problem is highly underdetermined, we demonstrate that, under the assumption that a predictor with small cross entropy loss exists, we can find a good predictor by coupling the marginal distributions of $X,Y_1,\dots ,Y_m$ in a manner that minimizes the conditional entropy $H(X|Y_1,\dots ,Y_m)$. This problem, which we call minimum conditional entropy coupling (MCEC), is a generalization of the widely studied minimum entropy coupling problem (where $m=1$). Although the MCEC problem is NP-hard, we give an algorithm with a time complexity that is polynomial in $N$, which gives a coupling with conditional entropy within $O(\log \log N)$ from the optimum, where $N$ is the dimension of the output coupling. We also evaluate the performance of the MCEC predictor on synthetic and real-world datasets.