On Learning and Testing Decision Tree

In this paper, we study learning and testing decision tree of size and depth that are significantly smaller than the number of attributes $n$. Our main result addresses the problem of poly$(n,1/\epsilon)$ time algorithms with poly$(s,1/\epsilon)$ query complexity (independent of $n$) that distinguish between functions that are decision trees of size $s$ from functions that are $\epsilon$-far from any decision tree of size $\phi(s,1/\epsilon)$, for some function $\phi > s$. The best known result is the recent one that follows from Blank, Lange and Tan,~\cite{BlancLT20}, that gives $\phi(s,1/\epsilon)=2^{O((\log^3s)/\epsilon^3)}$. In this paper, we give a new algorithm that achieves $\phi(s,1/\epsilon)=2^{O(\log^2 (s/\epsilon))}$. Moreover, we study the testability of depth-$d$ decision tree and give a {\it distribution free} tester that distinguishes between depth-$d$ decision tree and functions that are $\epsilon$-far from depth-$d^2$ decision tree. In particular, for decision trees of size $s$, the above result holds in the distribution-free model when the tree depth is $O(\log(s/\epsilon))$. We also give other new results in learning and testing of size-$s$ decision trees and depth-$d$ decision trees that follow from results in the literature and some results we prove in this paper.