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.
翻译:在本文中, 我们研究并测试大小和深度决定树, 这些树的大小和深度大大小于属性数 $美元。 我们的主要结果解决了美元( n, 1/\ epsilon) 美元时间算法问题, 其复杂性( 以美元为单位, 1/\ epsilon) 为单位( 以美元为单位) 。 在本文中, 我们给出了一个新的算法, 其大小为$( s, 1/\ eepsilon), 其规模大大小于属性数 $。 已知的最佳结果就是最近从 Blank, Lange 和 Tan, ⁇ cite { BlancLT20} 所推出的美元时间算法问题, 其复杂性为 美元( s, 1/\ epsilon) $( 美元), 其大小为美元( 美元) 大小决定树 的大小/ 值 值 值( i) 值( i) =2\ ligus2, levelop levelop $( $) $( $) $( $) lear) ) lear) 号决定的大小和 的决定值测试结果在树中, 我们在决定的深度中进行测试时, 的大小和 度分析结果在树的大小/ dereal- dealsestal_( estal_ 的大小和 测试结果, legreal_ lex 。