Binary Losses for Density Ratio Estimation

Estimating the ratio of two probability densities from a finite number of observations is a central machine learning problem. A common approach is to construct estimators using binary classifiers that distinguish observations from the two densities. However, the accuracy of these estimators depends on the choice of the binary loss function, raising the question of which loss function to choose based on desired error properties. For example, traditional loss functions, such as logistic or boosting loss, prioritize accurate estimation of small density ratio values over large ones, even though the latter are more critical in many applications. In this work, we start with prescribed error measures in a class of Bregman divergences and characterize all loss functions that result in density ratio estimators with small error. Our characterization extends results on composite binary losses from Reid & Williamson (2010) and their connection to density ratio estimation as identified by Menon & Ong (2016) . As a result, we obtain a simple recipe for constructing loss functions with certain properties, such as those that prioritize an accurate estimation of large density ratio values. Our novel loss functions outperform related approaches for resolving parameter choice issues of 11 deep domain adaptation algorithms in average performance across 484 real-world tasks including sensor signals, texts, and images. 2 Denote by y = 1 the event "drawn from P " and by y = -1 the event "drawn from Q". Then, the probability measure ρ on X × -1, 1 is defined by ρ(x|y = 1) := P (x), ρ(x|y = -1) := Q(x) and the marginal (w.r.t. y) Bernoulli measure ρY realizing y = 1 with probability 1 2 and y = -1 otherwise. It follows that β(x) = ρ(x|y=1) ρ(x|y=-1) = ρ(x,y=1)ρ Y (y=-1) ρ(x,y=-1)ρ Y (y=1) = ρ(x,y=1) ρ(x,y=-1) = ρ(x,y=1)ρ X (x) ρ(x,y=-1)ρ X (x) = ρ(y=1|x) ρ(y=-1|x) .