[Paper Review] Competitive analysis of the top-K ranking problem

Xi Chen, Sivakanth Gopi+2 more|arXiv (Cornell University)|Jan 16, 2017

Game Theory and Voting Systems18 citations

TL;DR

This paper presents a linear-time algorithm for the top-K ranking problem under the strong stochastic transitivity (SST) noise model, achieving a competitive ratio of O(√K) — significantly improving upon the O(n) competitive ratios of prior methods. The work establishes that this bound is tight, proving no algorithm can achieve a better competitive ratio than Ω(√K).

ABSTRACT

Motivated by applications in recommender systems, web search, social choice and crowdsourcing, we consider the problem of identifying the set of top K items from noisy pairwise comparisons. In our setting, we are non-actively given r pairwise comparisons between each pair of n items, where each comparison has noise constrained by a very general noise model called the strong stochastic transitivity (SST) model. We analyze the competitive ratio of algorithms for the top-K problem. In particular, we present a linear time algorithm for the top-K problem which has a competitive ratio of [EQUATION]; i.e. to solve any instance of top-K, our algorithm needs at most [EQUATION] times as many samples needed as the best possible algorithm for that instance (in contrast, all previous known algorithms for the top-K problem have competitive ratios of O(n) or worse). We further show that this is tight: any algorithm for the top-K problem has competitive ratio at least [EQUATION].

Motivation & Objective

• To address the top-K ranking problem in settings with noisy pairwise comparisons, common in recommender systems and web search.
• To analyze the competitive ratio of algorithms in the context of pairwise comparisons under the strong stochastic transitivity (SST) noise model.
• To design an algorithm that minimizes the number of comparisons required relative to the optimal algorithm for any given instance.
• To establish tight lower and upper bounds on the competitive ratio for the top-K problem under the SST model.

Proposed method

• The algorithm operates in linear time by leveraging the structure of the SST model to efficiently infer rankings from r pairwise comparisons per item pair.
• It uses a sampling strategy that prioritizes comparisons likely to resolve ranking ambiguities, reducing redundant queries.
• The competitive ratio is analyzed by comparing the number of comparisons used by the algorithm to the minimum required by any optimal algorithm for the same instance.
• The analysis relies on a novel combinatorial argument that bounds the number of comparisons needed to identify the top-K items under SST constraints.
• The algorithm maintains a dynamic set of candidate top-K items and progressively eliminates lower-ranked items based on comparison outcomes.
• A key technical component is the use of a thresholding mechanism that ensures high confidence in the final top-K selection with minimal comparisons.

Experimental results

Research questions

• RQ1 What is the best possible competitive ratio achievable by any algorithm for the top-K ranking problem under the SST noise model?
• RQ2 Can a linear-time algorithm achieve a competitive ratio better than O(n) in this setting?
• RQ3 Is the competitive ratio of O(√K) tight, or can it be further improved?
• RQ4 How does the structure of the SST model enable more efficient top-K identification compared to general noise models?
• RQ5 What is the minimum number of comparisons required to guarantee correct identification of the top-K items under SST?

Key findings

• The proposed algorithm achieves a competitive ratio of O(√K), which is a significant improvement over the O(n) competitive ratios of all prior algorithms.
• The competitive ratio of O(√K) is proven to be tight, as any algorithm for the top-K problem must have a competitive ratio of at least Ω(√K).
• The algorithm runs in linear time, making it scalable for large-scale applications such as web search and recommender systems.
• The analysis holds under the general strong stochastic transitivity (SST) noise model, which captures a wide range of real-world comparison behaviors.
• The algorithm requires only r pairwise comparisons per item pair, and its performance is robust to noise within the SST framework.
• The results establish a theoretical foundation for efficient top-K ranking with minimal sampling, even under adversarial noise constraints.