发表机构
Pennsylvania State University; Yale University（宾夕法尼亚州立大学; 耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究树数量趋向无穷时BART的行为，证明其收敛到特定高斯过程，引入随机树特征作为近似并证明其岭回归达极小极大最优学习率，兼具计算优势与更广适用性。
AI 中文摘要
贝叶斯加性回归树（Bayesian Additive Regression Trees, BART）在预测和因果推断问题中均展现出顶尖性能。过往理论研究尝试通过建立标准BART模型的后验收缩率来解释其优异表现，但这些收缩率高度依赖协变量数量。本文采用不同思路，研究树的数量趋向无穷时BART的行为，证明该 regime 下BART收敛到具有特定核的高斯过程（Gaussian Process, GP）。该核及其对应的再生核希尔伯特空间（RKHS）具备良好的推断特性，有助于解释BART的出色性能。本文引入随机树特征作为该极限GP的近似，并证明基于这些随机特征的岭回归达到极小极大最优学习率，该学习率仅与维度呈对数相关。除为BART的经验成功提供见解外，随机树特征相比传统MCMC估计还具备计算优势，且该随机特征近似使从业者可轻松将BART融入任何具有线性预测器的模型，拓展了BART的适用性与灵活性。
英文摘要
Bayesian Additive Regression Trees (BART) have shown state-of-the-art performance in both prediction and causal inference problems. Previous theoretical work has attempted to explain BART's superior performance by establishing posterior contraction rates for standard BART models, but these rates depend strongly on the number of covariates. Here, we take a different approach and study the behavior of BART as the number of trees grows towards infinity. We show that in this regime, BART converges to a Gaussian process (GP) with a particular kernel. The kernel and its corresponding reproducing kernel Hilbert space (RKHS) have favorable inferential properties that help explain BART's excellent performance. We introduce random tree features as an approximation to this limiting GP, and establish minimax-optimal learning rates for ridge regression on these random features that depend only logarithmically on dimension. In addition to providing insight into the empirical success of BART, random tree features offer a computational benefit over traditional MCMC estimation. The random-features approximation also allows practitioners to easily incorporate BART into any model which has a linear predictor, expanding the applicability and flexibility of BART.
Comments20 pages, 5 figures, plus references and appendices
