2024/11/07 by Yan Shuo Tan, Tan Yan, Tan, Yan Shuo +4 · 1 voice · 1 citation
Computer Science · Mathematics · #62G08 #68Q32 #Algorithms and Data Compression #Bayesian Methods and Mixture Models #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #G.3 #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Statistics Theory (math.ST) #cs.DS #cs.LG #math.ST #stat.ML
paper · pdf · doi:10.48550/arxiv.2411.04394
openalex publication_date 2024/11/07 · arxiv published 2024/11/07 · openalex created_date 2024/11/15 · arxiv updated 2025/09/10 · openalex updated_date 2026/07/28
Models based on recursive adaptive partitioning such as decision trees and their ensembles are popular for high-dimensional regression as they can potentially avoid the curse of dimensionality. Because empirical risk minimization (ERM) is computationally infeasible, these models are typically trained using greedy algorithms. Although effective in many cases, these algorithms have been empirically observed to get stuck at local optima. We explore this phenomenon in the context of learning sparse regression functions over d binary features, showing that when the true regression function f^* does not satisfy Abbe et al. (2022)'s Merged Staircase Property (MSP), greedy training requires exp(Ω(d)) to achieve low estimation error. Conversely, when f^* does satisfy MSP, greedy training can attain small estimation error with only O(log d) samples. This dichotomy mirrors that of two-layer neural networks trained with stochastic gradient descent (SGD) in the mean-field regime, thereby establishing a head-to-head comparison between SGD-trained neural networks and greedy recursive partitioning estimators. Furthermore, ERM-trained recursive partitioning estimators achieve low estimation error with O(log d) samples irrespective of whether f^* satisfies MSP, thereby demonstrating a statistical-computational trade-off for greedy training. Our proofs are based on a novel interpretation of greedy recursive partitioning using stochastic process theory and a coupling technique that may be of independent interest.