The Query Complexity of Certification

Abstract

We study the problem of {\sl certification}: given queries to a function $f:\{0,1{\}}^n\to \{0,1\}$ with certificate complexity $\le k$ and an input $x^{\star }$, output a size-$k$ certificate for $f$'s value on $x^{\star }$. This abstractly models a central problem in explainable machine learning, where we think of $f$ as a blackbox model that we seek to explain the predictions of. For monotone functions, a classic local search algorithm of Angluin accomplishes this task with $n$ queries, which we show is optimal for local search algorithms. Our main result is a new algorithm for certifying monotone functions with $O(k^8\log n)$ queries, which comes close to matching the information-theoretic lower bound of $\Omega (k\log n)$. The design and analysis of our algorithm are based on a new connection to threshold phenomena in monotone functions. We further prove exponential-in-$k$ lower bounds when $f$ is non-monotone, and when $f$ is monotone but the algorithm is only given random examples of $f$. These lower bounds show that assumptions on the structure of $f$ and query access to it are both necessary for the polynomial dependence on $k$ that we achieve.

30 pages, to appear in STOC'22. Edit: fixed typos and added references