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.

翻译：我们考虑了在训练数据不均匀的常见情况下，对分类树进行剪枝（即选择平衡偏差与方差的合适子树）的问题。具体而言，假设我们主要能够访问来自分布 $P_{X, Y}$ 的数据，而仅有少量来自目标分布 $Q_{X, Y}$ （其 $X$ 边际分布不同）的数据，我们提出了首个在此类场景下（当交叉验证及其他惩罚变体方法严重失效时）实现最优剪枝的高效流程。最优性是通过一种称为平均差异 $P_{X} \to Q_{X}$（在 $X$ 空间上取平均）的概念来推导的，该概念显著放松了近期被证明能严密刻画此类分布漂移下分类极限的“迁移指数”概念。这一放松的概念可被视为分布之间相对维度的度量，因为它与现有信息度量（如闵可夫斯基维数和Rényi维数）相关联。