Classification Tree Pruning Under Covariate Shift

We consider the problem of pruning a classification tree, that is, selecting a suitable subtree that balances bias and variance, in common situations with inhomogeneous training data. Namely, assuming access to mostly data from a distribution P-X,P-Y, but little data from a desired distribution Q(X,Y) with different X-marginals, we present the first efficient procedure for optimal pruning in such situations, when cross-validation and other penalized variants are grossly inadequate. Optimality is derived with respect to a notion of average discrepancy P-X -> Q(X) (averaged over X space) which significantly relaxes a recent notion-termed transfer-exponent-shown to tightly capture the limits of classification under such a distribution shift. Our relaxed notion can be viewed as a measure of relative dimension between distributions, as it relates to existing notions of information such as the Minkowski and Renyi dimensions.