[Paper Review] Ranking with Features: Algorithm and A Graph Theoretic Analysis

Aadirupa Saha, Arun Rajkumar|arXiv (Cornell University)|Aug 11, 2018

Data Management and Algorithms3 citations

TL;DR

This paper introduces the feature-Bradley-Terry-Luce (f-BTL) model and fBTL-LS algorithm to improve ranking from pairwise comparisons by leveraging item features. By modeling dependencies via feature representations, the method reduces sample complexity to O(α log α), where α is the number of independent items, significantly below the O(n log n) bound in standard BTL models.

ABSTRACT

We consider the problem of ranking a set of items from pairwise comparisons in the presence of features associated with the items. Recent works have established that $O(n\log(n))$ samples are needed to rank well when there is no feature information present. However, this might be sub-optimal in the presence of associated features. We introduce a new probabilistic preference model called feature-Bradley-Terry-Luce (f-BTL) model that generalizes the standard BTL model to incorporate feature information. We present a new least squares based algorithm called fBTL-LS which we show requires much lesser than $O(n\log(n))$ pairs to obtain a good ranking -- precisely our new sample complexity bound is of $O(α\log α)$, where $α$ denotes the number of `independent items' of the set, in general $α<< n$. Our analysis is novel and makes use of tools from classical graph matching theory to provide tighter bounds that sheds light on the true complexity of the ranking problem, capturing the item dependencies in terms of their feature representations. This was not possible with earlier matrix completion based tools used for this problem. We also prove an information theoretic lower bound on the required sample complexity for recovering the underlying ranking, which essentially shows the tightness of our proposed algorithms. The efficacy of our proposed algorithms are validated through extensive experimental evaluations on a variety of synthetic and real world datasets.

Motivation & Objective

• To address the inefficiency of existing ranking methods that require O(n log n) pairwise comparisons when item features are available.
• To model preference relationships in a way that captures dependencies among items through their feature representations.
• To develop a new algorithm that achieves better sample complexity by exploiting feature information.
• To provide a theoretical foundation for the improved sample complexity using graph matching theory.
• To establish tight information-theoretic lower bounds to validate the optimality of the proposed method.

Proposed method

• Proposes the feature-Bradley-Terry-Luce (f-BTL) model as a generalization of the standard BTL model to incorporate item features.
• Develops the fBTL-LS algorithm, a least squares-based method that leverages feature information to estimate preference scores.
• Applies tools from classical graph matching theory to derive tighter bounds on sample complexity.
• Uses the concept of 'independent items' (α) to characterize the true complexity of the ranking problem based on feature structure.
• Derives a sample complexity bound of O(α log α), where α ≪ n, reflecting reduced dependence on the total number of items.
• Proves an information-theoretic lower bound that confirms the tightness of the proposed sample complexity.

Experimental results

Research questions

• RQ1 Can we reduce the number of pairwise comparisons needed for accurate ranking by incorporating item features?
• RQ2 How does the structure of item features influence the sample complexity of ranking algorithms?
• RQ3 Can graph matching theory provide tighter bounds than matrix completion methods for preference learning?
• RQ4 Is the proposed fBTL-LS algorithm sample-optimal in the presence of feature information?
• RQ5 What is the fundamental limit (lower bound) on sample complexity for feature-augmented ranking?

Key findings

• The fBTL-LS algorithm achieves a sample complexity of O(α log α), where α is the number of independent items, significantly reducing the requirement compared to O(n log n) in standard models.
• The use of graph matching theory enables tighter bounds that capture item dependencies through feature representations, unlike prior matrix completion-based approaches.
• The proposed method demonstrates improved sample efficiency, especially when α ≪ n, due to effective exploitation of feature structure.
• An information-theoretic lower bound is established, showing that the O(α log α) sample complexity is asymptotically tight.
• Extensive experiments on synthetic and real-world datasets validate the superior performance and robustness of fBTL-LS across diverse settings.