2022/10/18 by Jean-Samuel Leboeuf, Leboeuf, Jean-Samuel, Frédéric LeBlanc +3 · 1 citation
Computer Science · #Bayesian Modeling and Causal Inference #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Neural Networks and Applications #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2210.10781
openalex publication_date 2022/10/18 · openalex created_date 2022/10/24 · openalex updated_date 2026/07/28
We revisit binary decision trees from the perspective of partitions of the data. We introduce the notion of partitioning function, and we relate it to the growth function and to the VC dimension. We consider three types of features: real-valued, categorical ordinal and categorical nominal, with different split rules for each. For each feature type, we upper bound the partitioning function of the class of decision stumps before extending the bounds to the class of general decision tree (of any fixed structure) using a recursive approach. Using these new results, we are able to find the exact VC dimension of decision stumps on examples of ℓ real-valued features, which is given by the largest integer d such that 2ℓ ≥ \binomd\lfloor(d)/(2)\rfloor. Furthermore, we show that the VC dimension of a binary tree structure with LT leaves on examples of ℓ real-valued features is in O(LT log(LTℓ)). Finally, we elaborate a pruning algorithm based on these results that performs better than the cost-complexity and reduced-error pruning algorithms on a number of data sets, with the advantage that no cross-validation is required.