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Top-$K$ ranking with a monotone adversary

Yuepeng Yang, Antares Chen, Lorenzo Orecchia, Cong Ma

TL;DR

This work studies top-$K$ ranking from pairwise comparisons under a semi-random monotone adversary, formalizing a semi-random BTL setting. It introduces a weighted maximum likelihood estimator (MLE) whose weights are computed via a fast SDP-based reweighting procedure, and provides a refined $\ell_\infty$ analysis that ties estimation accuracy to spectral properties of the weighted comparison graph. The authors show near-optimal sample complexity (up to a $\log^2 n$ factor) for exact top-$K$ recovery, with explicit bounds that depend on the spectral gap and degrees of the weighted graph, and they develop a nearly-linear time matrix-multiplicative-weight-update (MMWU) based SDP solver to compute the reweighting. Their approach integrates analytical advances with scalable algorithms, offering practical performance guarantees in semi-random, graph-based ranking scenarios and potential applicability to broader learning problems on perturbed graphs.

Abstract

In this paper, we address the top-$K$ ranking problem with a monotone adversary. We consider the scenario where a comparison graph is randomly generated and the adversary is allowed to add arbitrary edges. The statistician's goal is then to accurately identify the top-$K$ preferred items based on pairwise comparisons derived from this semi-random comparison graph. The main contribution of this paper is to develop a weighted maximum likelihood estimator (MLE) that achieves near-optimal sample complexity, up to a $\log^2(n)$ factor, where $n$ denotes the number of items under comparison. This is made possible through a combination of analytical and algorithmic innovations. On the analytical front, we provide a refined~$\ell_\infty$ error analysis of the weighted MLE that is more explicit and tighter than existing analyses. It relates the~$\ell_\infty$ error with the spectral properties of the weighted comparison graph. Motivated by this, our algorithmic innovation involves the development of an SDP-based approach to reweight the semi-random graph and meet specified spectral properties. Additionally, we propose a first-order method based on the Matrix Multiplicative Weight Update (MMWU) framework. This method efficiently solves the resulting SDP in nearly-linear time relative to the size of the semi-random comparison graph.

Top-$K$ ranking with a monotone adversary

TL;DR

This work studies top- ranking from pairwise comparisons under a semi-random monotone adversary, formalizing a semi-random BTL setting. It introduces a weighted maximum likelihood estimator (MLE) whose weights are computed via a fast SDP-based reweighting procedure, and provides a refined analysis that ties estimation accuracy to spectral properties of the weighted comparison graph. The authors show near-optimal sample complexity (up to a factor) for exact top- recovery, with explicit bounds that depend on the spectral gap and degrees of the weighted graph, and they develop a nearly-linear time matrix-multiplicative-weight-update (MMWU) based SDP solver to compute the reweighting. Their approach integrates analytical advances with scalable algorithms, offering practical performance guarantees in semi-random, graph-based ranking scenarios and potential applicability to broader learning problems on perturbed graphs.

Abstract

In this paper, we address the top- ranking problem with a monotone adversary. We consider the scenario where a comparison graph is randomly generated and the adversary is allowed to add arbitrary edges. The statistician's goal is then to accurately identify the top- preferred items based on pairwise comparisons derived from this semi-random comparison graph. The main contribution of this paper is to develop a weighted maximum likelihood estimator (MLE) that achieves near-optimal sample complexity, up to a factor, where denotes the number of items under comparison. This is made possible through a combination of analytical and algorithmic innovations. On the analytical front, we provide a refined~ error analysis of the weighted MLE that is more explicit and tighter than existing analyses. It relates the~ error with the spectral properties of the weighted comparison graph. Motivated by this, our algorithmic innovation involves the development of an SDP-based approach to reweight the semi-random graph and meet specified spectral properties. Additionally, we propose a first-order method based on the Matrix Multiplicative Weight Update (MMWU) framework. This method efficiently solves the resulting SDP in nearly-linear time relative to the size of the semi-random comparison graph.
Paper Structure (47 sections, 31 theorems, 167 equations, 3 figures, 2 algorithms)

This paper contains 47 sections, 31 theorems, 167 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Technical challenges
  3. Main contributions
  4. Analytical contributions.
  5. Algorithmic contributions.
  6. Related work
  7. Ranking with the BTL model.
  8. Semi-random adversary.
  9. Notation.
  10. Main results
  11. Problem formulation
  12. Semi-random comparison graph.
  13. Latent scores and pairwise comparisons.
  14. Top-$K$ recovery.
  15. Weighted MLE achieves exact recovery
  16. ...and 32 more sections

Key Result

Theorem 1

Suppose that $np\ge C_{1}\log (n)$ and $npL\ge C_{2}\kappa^{4}\log^3(n)$ for some large enough constants $C_{1},C_{2}>0$. With probability at least $1-n^{-10}$, Algorithm alg:main returns the weighted MLE $\widehat{\bm{\theta}}$ that obeys for some constant $C_{3}>0$. On this event, the top-$K$ items are recovered exactly as long as for some large enough constant $C_{4} > 0$. In addition, the re

Figures (3)

  • Figure 1: Adjacency matrix of a semi-random comparison graph. Each non-white square corresponds to a non-zero entry in the adjacency matrix.
  • Figure 2: Accuracy of top-$K$ recovery for MLE under uniform sampling and weighted MLE under semi-random sampling. See Section \ref{['app:experiments']} for the experiment setup.
  • Figure 3: Sample complexity required to exactly recover top-$K$ items v.s. score gap $\Delta_K$. The solid line represents the required sample complexity of the weighted MLE as given in Theorem \ref{['thm:weighted_MLE_semirandom']}. The dashed line represents the minimax lower bound given in Theorem \ref{['thm:minimax']}.

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2: Informal
  • Theorem 3
  • Corollary 1
  • proof
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 1: Conductance
  • ...and 33 more