Statistical and computational challenges in ranking
Abstract
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 $π^*$ of the experts - namely, $M_{π^{*-1}(i),k} \geq M_{π^{*-1}(i+1),k}$ for any $i\in [n-1], k \in [d]$. Then, ranking experts amounts to constructing an estimator of $π^*$. 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 $λ\mathbf 1\{S\times T\}$ 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.
Summary
This paper investigates the problem of ranking experts based on their answers to questions, modeled as a crowd-sourcing scenario. The core challenge lies in estimating the true ranking (permutation π*) of experts given a noisy matrix M, where M_{i,k} reflects the expected quality of expert i's answer to question k. The paper focuses on the "isotonic crowd-sourcing model," where M is assumed to be isotonic (non-increasing) up to an unknown permutation of the experts. The research aims to determine if statistically optimal ranking procedures can also be computationally efficient, specifically addressing the existence of "computational-statistical gaps." The paper starts by analyzing simpler sub-problems like sub-matrix detection and estimation, where the matrix M is constrained to have a specific block structure (λ1{S×T}). It provides an overview of existing results and proof techniques, focusing on computational lower bounds derived using low-degree polynomial methods. These simpler problems serve as a foundation for understanding the challenges in the general ranking problem. The authors then build upon these insights to discuss existing results and algorithmic ideas for the general isotonic ranking problem, presenting recent findings that disprove the existence of computational-statistical gaps in this model. The paper emphasizes the connection between the general ranking problem and a hierarchy of interacting sub-matrix estimation problems, highlighting how solving the former can be viewed as robustly solving several instances of the latter.
Key Insights
- •The paper demonstrates that the general isotonic ranking problem can be decomposed into a hierarchy of interacting sub-matrix estimation problems.
- •It presents recent results that *disprove* the existence of computational-statistical gaps in the general isotonic ranking model. This means that polynomial-time algorithms can achieve statistically optimal performance.
- •The analysis leverages low-degree polynomial methods to establish computational lower bounds for sub-matrix detection and estimation, serving as building blocks for the general ranking problem.
- •A novel proof technique for LD lower bounds in submatrix estimation is introduced, which is claimed to be more intuitive and transparent than usual techniques. This approach involves constructing an almost-orthonormal family of polynomials.
- •The paper provides explicit estimators (f_gs, f_rs, f_cs, f_ss,m) for submatrix detection and analyzes their performance in terms of error probabilities.
- •Lemma 2.1 establishes a connection between permuted isotonic matrices and submatrix models, showing that any isotonic matrix can be lower-bounded by a block matrix with comparable Frobenius norm (up to logarithmic factors).
- •The paper highlights the existence of a "detection-to-estimation gap" in some regimes of submatrix problems, where detecting the presence of a submatrix is easier than accurately estimating its location.
Practical Implications
- •The findings have implications for crowd-sourcing applications, where the goal is to accurately rank workers or experts based on their performance on tasks. The paper suggests that efficient algorithms can achieve optimal ranking accuracy.
- •The decomposition of the ranking problem into sub-matrix estimation problems can guide the development of hierarchical algorithms that focus on identifying and ranking subsets of experts based on their performance on specific questions.
- •Practitioners can use the presented estimators and the analysis of their performance to design ranking systems with provable guarantees on accuracy and computational efficiency.
- •The paper opens up future research directions in developing more sophisticated algorithms for sub-matrix estimation and exploring the connections between different computational models (e.g., low-degree polynomials, statistical queries).
- •The techniques developed in this paper could be applied to other ranking problems, such as ranking players in tournaments or sorting objects based on pairwise comparisons.