Quantitative Analysis of the Effect of Density Ratio Estimation in Covariate Shift Adaptation

TMLR Paper6061 Authors

Abstract:

In supervised learning, it is essential to assume that the test sample and the training sample come from the same distribution. But in reality, this assumption is frequently broken, which can lead to subpar performance from the learned model. We examine the learning problem under \emph{covariate shift}, in which the conditional distribution of labels given covariates does not change despite the covariate distribution shifting. Two-step procedures, which first compute the density ratio and then carry out importance-weighted empirical risk minimization, are a popular family of methods for addressing covariate shift. However, the two-step techniques' performance could degrade due to estimation error of the density ratio.
Unfortunately, the extent of the density ratio estimation error that affects the accuracy of learning algorithms is rarely analyzed. This paper accordingly provides a quantitative answer to this question. Specifically, we formulate the two-step covariate adaptation methods as a meta-algorithm. We show that the effect of the density ratio estimation error on the excess risk bound of the meta algorithm is of the fourth order, i.e., $\mathcal{O}\left(\epsilon_{1}\left(\mathcal{G}, S_{s1}, S_t, \delta/2\right)^4\right)$, if the true risk satisfies a requirement known as the \emph{derivative vanishing} property, where $\epsilon_{1}\left(\mathcal{G}, S_{s1}, S_t, \delta/2\right)$ is the convergence rate of the density ratio estimation algorithm, $\mathcal{G}$ is the density ratio function class, $S_{s1}$ and $S_t$ are the samples generated by training distribution and test distribution respectively, and $\delta/2$ is the confidence parameter. Moreover, we analyze the impact of two specific density ratio estimation algorithms, Kullback-Leibler Importance Estimation Procedure and Kernel unconstrained Least-Squares Importance Fitting, on the final classifier's generalization error. We also report the experimental results of two-step covariate shift adaptation with a toy classification dataset using KLIEP.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Matthew_J._Holland1

Submission Number: 6061