具有单调对手的Top-$K$排序
Top-$K$ ranking with a monotone adversary
AI总结：
针对带单调对手的Top-K排序问题，提出加权极大似然估计器，结合SDP图重加权与MMWU一阶方法，实现近最优样本复杂度，误差分析更精细且算法效率近线性。
摘要：
本文研究具有单调对手的Top-$K$排序问题。我们考虑比较图随机生成、且对手可添加任意边的场景，统计学家的目标是基于该半随机比较图得到的成对比较结果，准确识别出最受偏好的前$K$个物品。本文的主要贡献是提出一种加权极大似然估计器（MLE），其样本复杂度达到近最优水平，仅相差一个$\log^2(n)$因子，其中$n$表示待比较的物品总数。这一结果得益于分析层面与算法层面的创新结合。在分析层面，我们对加权极大似然估计器的$\ell_\infty$误差进行了更精细的分析，相比现有分析结果更明确、更紧致，该分析将$\ell_\infty$误差与加权比较图的谱性质关联起来。受此启发，我们的算法创新在于提出一种基于半定规划（SDP）的方法，用于对半随机图进行重加权，使其满足指定的谱性质。此外，我们还提出了一种基于矩阵乘法权重更新（MMWU）框架的一阶方法，该方法能以相对于半随机比较图规模近线性的时间，高效求解得到的半定规划问题。
英文摘要：
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.
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