Classification Tree Pruning Under Covariate Shift
This addresses a practical issue for machine learning practitioners dealing with distribution shifts, but it is incremental as it builds on existing transfer-exponent concepts.
The paper tackles the problem of pruning classification trees under covariate shift, where training data comes from a different distribution than the target, by introducing an efficient procedure for optimal pruning based on a relaxed average discrepancy measure, achieving optimality in this setting.
We consider the problem of \emph{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 $P_{X, Y}$, 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 \emph{average discrepancy} $P_{X} \to Q_{X}$ (averaged over $X$ space) which significantly relaxes a recent notion -- termed \emph{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 \emph{relative dimension} between distributions, as it relates to existing notions of information such as the Minkowski and Renyi dimensions.