Optimal level set estimation for non-parametric tournament and crowdsourcing problems

• URL: http://arxiv.org/abs/2408.15356v1
• Date: Tue, 27 Aug 2024 18:28:31 GMT
• Title: Optimal level set estimation for non-parametric tournament and crowdsourcing problems
• Authors: Maximilian Graf, Alexandra Carpentier, Nicolas Verzelen,
• Abstract summary: Motivated by crowdsourcing, we consider a problem where we partially observe the correctness of the answers of $n$ experts on $d$ questions. In this paper, we assume that the matrix $M$ containing the probability that expert $i$ answers correctly to question $j$ is bi-isotonic up to a permutation of it rows and columns. We construct an efficient-time algorithm that turns out to be minimax optimal for this classification problem.
• Score: 49.75262185577198
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Motivated by crowdsourcing, we consider a problem where we partially observe the correctness of the answers of $n$ experts on $d$ questions. In this paper, we assume that both the experts and the questions can be ordered, namely that the matrix $M$ containing the probability that expert $i$ answers correctly to question $j$ is bi-isotonic up to a permutation of it rows and columns. When $n=d$, this also encompasses the strongly stochastic transitive (SST) model from the tournament literature. Here, we focus on the relevant problem of deciphering small entries of $M$ from large entries of $M$, which is key in crowdsourcing for efficient allocation of workers to questions. More precisely, we aim at recovering a (or several) level set $p$ of the matrix up to a precision $h$, namely recovering resp. the sets of positions $(i,j)$ in $M$ such that $M_{ij}>p+h$ and $M_{i,j}<p-h$. We consider, as a loss measure, the number of misclassified entries. As our main result, we construct an efficient polynomial-time algorithm that turns out to be minimax optimal for this classification problem. This heavily contrasts with existing literature in the SST model where, for the stronger reconstruction loss, statistical-computational gaps have been conjectured. More generally, this shades light on the nature of statistical-computational gaps for permutations models.

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