Learning decision trees from random examples

Abstract

We define the rank of a decision tree and show that for any fixed r , the class of all decision trees of rank at most r on n Boolean variables is learnable from random examples in time polynomial in n and linear in 1/ɛ and log(1/δ), where ɛ is the accuracy parameter and δ is the confidence parameter. Using a suitable encoding of variables, Rivest's polynomial learnability result for decision lists can be interpreted as a special case of this result for rank 1. As another corollary, we show that decision trees on n Boolean variables of size polynomial in n are learnable from random examples in time linear in n O (log n ) , 1/ɛ, and log(1/δ). As a third corollary, we show that Boolean functions that have polynomial size DNF expressions for both their positive and their negative instances are learnable from random examples in time linear in n O ((log n ) 2 ) , 1/ɛ, and log(1/δ).