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# Universal guarantees for decision tree induction via a higher-order splitting criterion

NIPS 2020, (2020)

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摘要

We propose a simple extension of top-down decision tree learning heuristics such as ID3, C4.5, and CART. Our algorithm achieves provable guarantees for all target functions $f: \{-1,1\}^n \to \{-1,1\}$ with respect to the uniform distribution, circumventing impossibility results showing that existing heuristics fare poorly even for simp...更多

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简介

- Decision trees have long been a workhorse of machine learning. Their simple hierarchical structure makes them appealing in terms of interpretability and explanatory power.
- The authors design a simple top-down decision tree learning algorithm based on the new splitting criterion (⋆), and show that it achieves provable guarantees for all target functions f : {±1}n → {±1}, i.e. a universal guarantee of the form (♦).
- Given a partial tree T ◦ and a target function f : {±1}n → {±1}, there is a canonical completion of T ◦ that minimizes classification error errorf (T ): label every leaf l of T ◦ with sign(E[fl]), where fl denotes the subfunction of f obtained by restricting its attributes according to the path that leads to l.

重点内容

- Decision trees have long been a workhorse of machine learning
- We depart from the overall theme of prior works: instead of analyzing the performance of impuritybased heuristics when run on restricted classes of target functions, we propose a simple and efficient extension of these heuristics, and show that it achieves the sought-for universal guarantee (♦)
- We design a simple top-down decision tree learning algorithm based on our new splitting criterion (⋆), and show that it achieves provable guarantees for all target functions f : {±1}n → {±1}, i.e. a universal guarantee of the form (♦)
- Theorem 1 should be contrasted with the impossibility result discussed in the introduction, which shows that there are very simple target functions for which opt4 = 0, and yet any impuritybased heuristic may build a complete tree of size Ω(2n) (in time Ω(2n)) before achieving any non-trivial error
- We will show that Theorem 1 follows as a consequence of the following more general result, which gives a performance guarantee for BuildStabilizingDT that scales according to the noise sensitivity of the target function f
- We have designed a new top-down decision tree learning algorithm that achieves provable guarantees for all target functions f : {±1}n → {±1} with respect to the uniform distribution. This circumvents impossibility results showing that such a universal guarantee is provably unachievable by existing top-down heuristics such as ID3, C4.5, and CART

结果

- Theorem 1 should be contrasted with the impossibility result discussed in the introduction, which shows that there are very simple target functions for which opt4 = 0, and yet any impuritybased heuristic may build a complete tree of size Ω(2n) (in time Ω(2n)) before achieving any non-trivial error
- Recent work of Blanc, Lange, and Tan [BLT20b] gives a top-down algorithm for learning decision trees that achieves provable guarantees for all target functions f .
- Fact 3.1 motivates the following question: given a partial decision tree T ◦ and a target function f , is there a leaf l⋆
- The authors will show that Theorem 1 follows as a consequence of the following more general result, which gives a performance guarantee for BuildStabilizingDT that scales according to the noise sensitivity of the target function f .
- T}, the authors write Tj◦ to denote the size-j partial decision tree that BuildStabilizingDT constructs after j − 1 iterations.
- The authors will use the potential function, the noise sensitivity of f with respect to a partial decision tree T ◦ (Definition 4), to bound this number.
- The authors have designed a new top-down decision tree learning algorithm that achieves provable guarantees for all target functions f : {±1}n → {±1} with respect to the uniform distribution.

结论

- The analysis of the algorithm draws on Fourier-analytic techniques, and is based on a view of the top-down decision tree construction process as a noise stabilization procedure.
- The focus of the work has been on achieving a universal guarantee: the new algorithm constructs, for all target functions f and size parameters s, a decision tree hypothesis of quasipoly(s) size that achieves error close to that of the best size-s decision tree f .
- Are there broad and natural subclasses of target functions for which the quasipoly(s) bound on the size of the decision tree hypothesis can be improved to poly(s), or even O(s)?

基金

- Guy, Jane, and Li-Yang were supported by NSF award CCF-192179 and NSF CAREER award CCF-1942123
- Neha was supported by NSF award 1704417

引用论文

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