CART算法的停止规则与空间适应性

中文摘要 AI 辅助

流行的回归树CART算法将贪心分裂规则与停止规则相结合，但分裂规则已得到充分研究，而停止规则的统计作用却鲜为人知。与此同时，尽管通过贝叶斯方法或经验风险最小化（ERM）拟合的回归树已被证明能对局部平滑性和各向异性实现空间自适应，但CART是否能达到相同的自适应效果尚不明确。我们针对这些空白展开研究，证明在回归函数与协变量分布具有空间异质性和各向异性平滑性及适当结构假设的条件下，采用最小不纯度下降（MID）停止规则及合适阈值的CART，可实现逐点速率，该速率在对数因子范围内达到极小极大最优，且此速率在整个定义域内所有点上同时成立。此外，我们证明采用广泛使用的最小叶节点大小停止规则无法实现空间自适应。综上，这些结果明确了MID停止规则的精确统计作用，并为CART的经验成功提供了理论依据。

英文摘要

The popular CART algorithm for regression trees combines a greedy splitting rule with a stopping rule, but while the splitting rule has been well studied, the statistical role of stopping rules is less well understood. Meanwhile, although regression trees fit using Bayesian methods or via empirical risk minimization (ERM) have been shown to be spatially adaptive to local smoothness and anisotropy, it is unknown whether CART can achieve the same adaptation. We address these gaps by proving that, under spatially heterogeneous and anisotropic smoothness and appropriate structural assumptions on the regression function and covariate distribution, CART with the minimum impurity decrease (MID) stopping rule and a suitable threshold achieves pointwise rates that are minimax up to logarithmic factors. These rates hold simultaneously over all points in the domain. Moreover, we prove that spatial adaptation cannot be achieved under the widely used minimum leaf size stopping rule. Together, these results establish a precise statistical role for the MID stopping rule and provide a theoretical basis for the empirical success of CART.

发表机构

• National University of Singapore（新加坡国立大学）

机构由 AI 辅助整理，请以论文原文为准。