Inapproximability of VC Dimension and Littlestone's Dimension

Abstract

We study the complexity of computing the VC Dimension and Littlestone's Dimension. Given an explicit description of a finite universe and a concept class (a binary matrix whose $(x,C)$-th entry is $1$ iff element $x$ belongs to concept $C$), both can be computed exactly in quasi-polynomial time ($n^{O(\log n)}$). Assuming the randomized Exponential Time Hypothesis (ETH), we prove nearly matching lower bounds on the running time, that hold even for approximation algorithms.

To appear in Conference On Learning Theory (COLT) 2017

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• Complexity theoretic limitations on learning DNF's
• Detecting communities is hard, and counting them is even harder