Dynamic Ranking with the BTL Model: A Nearest Neighbor based Rank Centrality Method

Abstract

Many applications such as recommendation systems or sports tournaments involve pairwise comparisons within a collection of n items, the goal being to aggregate the binary outcomes of the comparisons in order to recover the latent strength and/or global ranking of the items. In recent years, this problem has received significant interest from a theoretical perspective with a number of methods being proposed, along with associated statistical guarantees under the assumption of a suitable generative model.While these results typically collect the pairwise comparisons as one comparison graph G, however in many applications -such as the outcomes of soccer matches during a tournamentthe nature of pairwise outcomes can evolve with time. Theoretical results for such a dynamic setting are relatively limited compared to the aforementioned static setting. We study in this paper an extension of the classic BTL (Bradley-Terry-Luce) model for the static setting to our dynamic setup under the assumption that the probabilities of the pairwise outcomes evolve smoothly over the time domain [0, 1]. Given a sequence of comparison graphs (G t ) t ∈T on a regular grid T ⊂ [0, 1], we aim at recovering the latent strengths of the items w t ∈ R n at any time t ∈ [0, 1]. To this end, we adapt the Rank Centrality method -a popular spectral approach for ranking in the static case -by locally averaging the available data on a suitable neighborhood of t. When (G t ) t ∈T is a sequence of Erdös-Renyi graphs, we provide non-asymptotic 2 and ∞ error bounds for estimating w * t which in particular establishes the consistency of this method in terms of n, and the grid size |T |. We also complement our theoretical analysis with experiments on real and synthetic data.

Detecting Abrupt Changes in Sequential Pairwise Comparison Data

“…Ranking analysis with temporal variants has also become increasingly important because of the growing needs for models and methods to handle time-dependent data. A series of results in this direction can be found in Glickman (1993), Glickman and Stern (1998), Cattelan et al (2013), Lopez et al (2018), Maystre et al (2019), Bong et al (2020), Karlé and Tyagi (2021) and references therein. Much of the aforementioned literature on time-varying BTL model postulates that temporal changes in the model parameters are smooth functions of time and thus occur gradually on a relatively large time scale.…”

Dynamic Ranking and Translation Synchronization

“…• Recently, [24,2] considered a dynamic BTL model where the parameters of the model evolve in a Lipschitz smooth manner. For this setting, pointwise consistency rates were derived for estimating the strength vectors at any given time t ∈ [0, 1] -this was shown for a nearest neighbor based rank centrality method in [24], and for a maximum likelihood based approach in [2]. Unlike the local smoothness assumptions made in these works, we consider a global smoothness assumption, akin to the quadratic variation of a function over a grid.…”

“…• We consider a dynamic version of the TranSync model, where the latent strength parameters evolve smoothly with time. The smoothness assumption we work under is global in nature, as opposed to local assumptions previously considered in the literature [24,2]. We propose two estimators for this model, one based on a smoothness-penalized least squares approach and the other based on a projection method, which can be considered as extensions of the Laplacian smoothing and Laplacian eigenmaps estimators [37], respectively.…”

“…A well-studied model in ranking problems is the BTL model [3], and it has recently been extended to a dynamic setting [2,24]. It posits that at each time t, the probability that an item is preferred to another item only depends on their strengths.…”

“…The ground truth w * = exp(z * ) is generated similarly as in the Dynamic TranSync setup, as well as the observation graphs. However the measurements y ij (t) are now generated according to the Dynamic BTL model (see [24] for more details). In this particular setup, we will compare the performance of our methods with two other approaches that focus on dynamic ranking for the BTL model, namely, the Maximum-Likelihood Estimation (MLE) method [2] and the Dynamic Rank Centrality (DRC) method [24].…”