Open Problem: Properly learning decision trees in polynomial time?

TL;DR

This paper discusses the open challenge of developing a polynomial-time algorithm for properly learning decision trees, reviewing recent progress and outlining future research directions.

Contribution

It highlights the open problem, reviews recent algorithms, and proposes intermediate milestones towards polynomial-time learning of decision trees.

Findings

Recent algorithm runs in $n^{O(\log\log n)}$ time

Previous best algorithm runs in $n^{O(\log n)}$ time

Open problem of polynomial-time learning remains unsolved

Abstract

The authors recently gave an $n^{O(\log\log n)}$ time membership query algorithm for properly learning decision trees under the uniform distribution (Blanc et al., 2021). The previous fastest algorithm for this problem ran in $n^{O(\log n)}$ time, a consequence of Ehrenfeucht and Haussler (1989)'s classic algorithm for the distribution-free setting. In this article we highlight the natural open problem of obtaining a polynomial-time algorithm, discuss possible avenues towards obtaining it, and state intermediate milestones that we believe are of independent interest.