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 informationtheoretic 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 exponentialin$k$ lower bounds when $f$ is nonmonotone, 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 DOI:
 10.48550/arXiv.2201.07736
 arXiv:
 arXiv:2201.07736
 Bibcode:
 2022arXiv220107736B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity
 EPrint:
 30 pages, to appear in STOC'22. Edit: fixed typos and added references