Adaptive estimation of multivariate piecewise polynomials and bounded variation functions by optimal decision trees
用最优决策树自适应估计多元分段多项式与有界变差函数AI 解读 ↓
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用最优决策树自适应估计多元分段多项式与有界变差函数
- 研究问题
- 如何在一般维度的固定设计回归中，以计算可行的最优决策树估计分段光滑函数，并自适应于不同函数复杂度？
- 方法与思路
- 作者扩展Dyadic CART并研究最优回归树，用分段多项式拟合，通过动态规划计算，再推导有限样本风险的预言不等式。
- 材料陈述的贡献
- 摘要称，风险界覆盖分段多项式函数类，并揭示有界变差函数上的极小极大率最优性、自适应性及特定比较优势。
- 阅读关注点
- 应核对矩形分区的规则性、树优化的计算复杂度、风险界条件，以及与总变差去噪和趋势滤波比较的具体范围。
- 阅读边界
- 依据版本未注明的OpenAlex索引摘要；所给风险界公式排版不清，须查原PDF，不能重构公式或推断普遍优越性。
原文摘要
Proposed by Donoho (Ann. Statist. 25 (1997) 1870–1911), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article, we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the context of estimating piecewise smooth functions in general dimensions in the fixed design setup. More precisely, these optimal decision tree estimators fit piecewise polynomials of any given degree. Like Dyadic CART in two dimensions, we reason that these estimators can also be computed in polynomial time in the sample size N via dynamic programming. We prove oracle inequalities for the finite sample risk of Dyadic CART and ORT, which imply tight risk bounds for several function classes of interest. First, they imply that the finite sample risk of ORT of order r≥0 is always bounded by CklogN N whenever the regression function is piecewise polynomial of degree r on some reasonably regular axis aligned rectangular partition of the domain with at most k rectangles. Beyond the univariate case, such guarantees are scarcely available in the literature for computationally efficient estimators. Second, our oracle inequalities uncover minimax rate optimality and adaptivity of the Dyadic CART estimator for function spaces with bounded variation. We consider two function spaces of recent interest where multivariate total variation denoising and univariate trend filtering are the state of the art methods. We show that Dyadic CART enjoys certain advantages over these estimators while still maintaining all their known guarantees.