We prove that it is NP-hard to properly PAC learn decision trees with queries, resolving a longstanding open problem in learning theory (Bshouty 1993; Guijarro-Lavin-Raghavan 1999; Mehta-Raghavan 2002; Feldman 2016). While there has been a long line of work, dating back to (Pitt-Valiant 1988), establishing the hardness of properly learning decision trees from random examples, the more challenging setting of query learners necessitates different techniques and there were no previous lower bounds. En route to our main result, we simplify and strengthen the best known lower bounds for a different problem of Decision Tree Minimization (Zantema-Bodlaender 2000; Sieling 2003).
On a technical level, we introduce the notion of hardness distillation, which we study for decision tree complexity but can be considered for any complexity measure: for a function that requires large decision trees, we give a general method for identifying a small set of inputs that is responsible for its complexity. Our technique even rules out query learners that are allowed constant error. This contrasts with existing lower bounds for the setting of random examples which only hold for inverse-polynomial error.
Our result, taken together with a recent almost-polynomial time query algorithm for properly learning decision trees under the uniform distribution (Blanc-Lange-Qiao-Tan 2022), demonstrates the dramatic impact of distributional assumptions on the problem.
Institute for Data, Econometrics, Algorithms, and Learning (IDEAL)（数据、计量经济学、算法和学习研究所）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Weizmann Institute of Science（魏茨曼科学研究院）
Preventing False Discovery in Interactive Data Analysis is Hard
IBM Research Almaden（IBM阿尔马登研究院）；Harvard University School of Engineering and Applied Sciences（哈佛大学工程与应用科学学院）；Center for Research on Computation and Society（计算与社会研究中心）
