Caleb Koch

Caleb Koch, Carmen Strassle, Li-Yang Tan

We consider the task of properly PAC learning decision trees with queries. Recent work of Koch, Strassle, and Tan showed that the strictest version of this task, where the hypothesis tree $T$ is required to be optimally small, is NP-hard. Their work leaves open the question of whether the task remains intractable if $T$ is only required to be close to optimal, say within a factor of 2, rather than exactly optimal. We answer this affirmatively and show that the task indeed remains NP-hard even if $T$ is allowed to be within any constant factor of optimal. More generally, our result allows for a smooth tradeoff between the hardness assumption and the inapproximability factor. As Koch et al.'s techniques do not appear to be amenable to such a strengthening, we first recover their result with a new and simpler proof, which we couple with a new XOR lemma for decision trees. While there is a large body of work on XOR lemmas for decision trees, our setting necessitates parameters that are extremely sharp, and are not known to be attainable by existing XOR lemmas. Our work also carries new implications for the related problem of Decision Tree Minimization.

Caleb Koch, Carmen Strassle, Li-Yang Tan

We connect the problem of properly PAC learning decision trees to the parameterized Nearest Codeword Problem ($k$-NCP). Despite significant effort by the respective communities, algorithmic progress on both problems has been stuck: the fastest known algorithm for the former runs in quasipolynomial time (Ehrenfeucht and Haussler 1989) and the best known approximation ratio for the latter is $O(n/\log n)$ (Berman and Karpinsky 2002; Alon, Panigrahy, and Yekhanin 2009). Research on both problems has thus far proceeded independently with no known connections. We show that $\textit{any}$ improvement of Ehrenfeucht and Haussler's algorithm will yield $O(\log n)$-approximation algorithms for $k$-NCP, an exponential improvement of the current state of the art. This can be interpreted either as a new avenue for designing algorithms for $k$-NCP, or as one for establishing the optimality of Ehrenfeucht and Haussler's algorithm. Furthermore, our reduction along with existing inapproximability results for $k$-NCP already rule out polynomial-time algorithms for properly learning decision trees. A notable aspect of our hardness results is that they hold even in the setting of $\textit{weak}$ learning whereas prior ones were limited to the setting of strong learning.

Lorenzo Beretta, Nathaniel Harms, Caleb Koch

For a function $f \colon \{0,1\}^n \to \{0,1\}$, the junta testing problem asks whether $f$ depends on only $k$ variables. If $f$ depends on only $k$ variables, the feature selection problem asks to find those variables. We prove that these two tasks are statistically equivalent. Specifically, we show that the ``brute-force'' algorithm, which checks for any set of $k$ variables consistent with the sample, is simultaneously sample-optimal for both problems, and the optimal sample size is \[ Θ\left(\frac 1 \varepsilon \left( \sqrt{2^k \log {n \choose k}} + \log {n \choose k}\right)\right). \]