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Zusammenfassung:
Assume we are given sets of observations of training and test data, where (unlike in the classical setting) the training and test distributions are allowed to differ. Thus for learning purposes, we face the problem of re-weighting the training data such that its distribution more closely matches that of the test data. We consider specifically the case where the difference in training and test distributions occurs only in the marginal distribution of the covariates: the conditional distribution of the outputs given the covariates is unchanged. We achieve covariate shift correction by matching covariate distributions between training and test sets in a high dimensional feature space (specifically, a reproducing kernel Hilbert space). This approach does not require distribution estimation, making it suited to high dimensions and structured data, where distribution estimates may not be practical.
We first describe the general setting of covariate shift correction, and the importance weighting approach. While direct density estimation provides an estimate of the importance weights, this has two potential disadvantages: it may not offer the best bias/variance tradeoff, and density estimation might be difficult on complex, high dimensional domains (such as text). We then describe how distributions may be mapped to reproducing kernel Hilbert spaces (RKHS), and review distances between such mappings. We demonstrate a transfer learning algorithm that reweights the training points such that their RKHS mapping matches that of the (unlabeled) test points. The sample weights are obtained by a simple quadratic programming procedure. Our correction method yields its greatest and most consistent advantages when the learning algorithm returns a classifier/regressor that is "simpler" than the data might suggest. On the other hand, even an ideal sample reweighting may not be of practical benefit given a sufficiently powerful learning algorithm (if available).
