Statistical and computational challenges in ranking

We consider the problem of ranking $n$ experts according to their abilities, based on the correctness of their answers to $d$ questions. This is modeled by the so-called crowd-sourcing model, where the answer of expert $i$ on question $k$ is modeled by a random entry, parametrized by $M_{i,k}$ which is increasing linearly with the expected quality of the answer. To enable the unambiguous ranking of the experts by ability, several assumptions on $M$ are available in the literature. We consider here the general isotonic crowd-sourcing model, where $M$ is assumed to be isotonic up to an unknown permutation $\pi^*$ of the experts - namely, $M_{\pi^{*-1}(i),k} \geq M_{\pi^{*-1}(i+1),k}$ for any $i\in [n-1], k \in [d]$. Then, ranking experts amounts to constructing an estimator of $\pi^*$. In particular, we investigate here the existence of statistically optimal and computationally efficient procedures and we describe recent results that disprove the existence of computational-statistical gaps for this problem. To provide insights on the key ideas, we start by discussing simpler and yet related sub-problems, namely sub-matrix detection and estimation. This corresponds to specific instances of the ranking problem where the matrix $M$ is constrained to be of the form $\lambda \mathbf 1\{S\times T\}$ where $S\subset [n], T\subset [d]$. This model has been extensively studied. We provide an overview of the results and proof techniques for this problem with a particular emphasis on the computational lower bounds based on low-degree polynomial methods. Then, we build upon this instrumental sub-problem to discuss existing results and algorithmic ideas for the general ranking problem.