MINIMAX OPTIMAL RATES FOR MONDRIAN TREES AND FORESTS

• 成果类型: Article
• 署名作者: Mourtada, Jaouad; Gaiffas, Stephane; Scornet, Erwan
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; Universite Paris Cite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1886
• 发表日期: 2020
• 页码: 2253-2276
• 关键词: Classification

摘要

Introduced by Breiman (Mach. Learn. 45 (2001) 5-32), Random Forests are widely used classification and regression algorithms. While being initially designed as batch algorithms, several variants have been proposed to handle online learning. One particular instance of such forests is the Mondrian forest (In Adv. Neural Inf. Process. Syst. (2014) 3140-3148; In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AIS-TATS) (2016)), whose trees are built using the so-called Mondrian process, therefore allowing to easily update their construction in a streaming fashion. In this paper we provide a thorough theoretical study of Mondrian forests in a batch learning setting, based on new results about Mondrian partitions. Our results include consistency and convergence rates for Mondrian trees and forests, that turn out to be minimax optimal on the set of s-Holder function with s is an element of (0, 1] (for trees and forests) and s is an element of (1, 2] (for forests only), assuming a proper tuning of their complexity parameter in both cases. Furthermore, we prove that an adaptive procedure (to the unknown s is an element of (0, 2]) can be constructed by combining Mondrian forests with a standard model aggregation algorithm. These results are the first demonstrating that some particular random forests achieve minimax rates in arbitrary dimension. Owing to their remarkably simple distributional properties, which lead to minimax rates, Mondrian trees are a promising basis for more sophisticated yet theoretically sound random forests variants.