Reconstructing decision trees

TL;DR

This paper introduces the first reconstruction algorithm for decision trees, enabling approximation of a target function with a decision tree of controlled size and error, significantly improving testing efficiency for decision tree-related properties.

Contribution

The paper presents a novel reconstruction algorithm for decision trees that achieves near-optimal size and error bounds with efficient query complexity, advancing decision tree property testing.

Findings

Reconstruction algorithm produces a decision tree close to the original function.

Efficient tolerant tester distinguishes functions near and far from size-s decision trees.

Polylogarithmic dependence on size s improves over previous testers.

Abstract

We give the first {\sl reconstruction algorithm} for decision trees: given queries to a function $f$ that is $\mathrm{opt}$-close to a size-$s$ decision tree, our algorithm provides query access to a decision tree $T$ where: $\circ$ $T$ has size $S = s^{O((\log s)^2/\varepsilon^3)}$; $\circ$ $\mathrm{dist}(f,T)\le O(\mathrm{opt})+\varepsilon$; $\circ$ Every query to $T$ is answered with $\mathrm{poly}((\log s)/\varepsilon)\cdot \log n$ queries to $f$ and in $\mathrm{poly}((\log s)/\varepsilon)\cdot n\log n$ time. This yields a {\sl tolerant tester} that distinguishes functions that are close to size-$s$ decision trees from those that are far from size-$S$ decision trees. The polylogarithmic dependence on $s$ in the efficiency of our tester is exponentially smaller than that of existing testers. Since decision tree complexity is well known to be related to numerous other boolean function properties, our results also provide a new algorithms for reconstructing and testing these properties.