  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.
