适应光滑性和稀疏性的贝叶斯回归树集成
Bayesian Regression Tree Ensembles that Adapt to Smoothness and Sparsity
摘要：
决策树集成是一个有用的工具，用于获得回归函数的灵活估计。这些方法的示例包括梯度提升决策树、随机森林和贝叶斯CART。树集合的两个潜在缺点是它们缺乏平滑性和易受维数灾难的影响。我们表明，这些问题可以克服，而不是考虑稀疏诱导软决策树中的决定被视为概率。我们在贝叶斯加性回归树框架的背景下实现了这一点，并通过在基准数据集上的测试说明了其良好的性能。我们提供了强有力的理论支持，我们的方法表明，后验分布集中在极小极大率（最多对数因子）的稀疏函数和功能与添加剂的结构在高维政权的协变量空间的维数允许增长近指数的样本量。我们的方法也适应未知的平滑度和稀疏度，并可以通过对现有的BART算法进行最小的修改来实现。
英文摘要：
Ensembles of decision trees are a useful tool for obtaining for obtaining flexible estimates of regression functions. Examples of these methods include gradient boosted decision trees, random forests, and Bayesian CART. Two potential shortcomings of tree ensembles are their lack of smoothness and vulnerability to the curse of dimensionality. We show that these issues can be overcome by instead considering sparsity inducing soft decision trees in which the decisions are treated as probabilistic. We implement this in the context of the Bayesian additive regression trees framework, and illustrate its promising performance through testing on benchmark datasets. We provide strong theoretical support for our methodology by showing that the posterior distribution concentrates at the minimax rate (up-to a logarithmic factor) for sparse functions and functions with additive structures in the high-dimensional regime where the dimensionality of the covariate space is allowed to grow near exponentially in the sample size. Our method also adapts to the unknown smoothness and sparsity levels, and can be implemented by making minimal modifications to existing BART algorithms.
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