Top-$K$ ranking with a monotone adversary
<|reference_start|>Top-$K$ ranking with a monotone adversary: 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.<|reference_end|>
@article{yang2024top-$k$,
    title={Top-$K$ ranking with a monotone adversary},
    author={Yuepeng Yang, Antares Chen, Lorenzo Orecchia, Cong Ma},
    journal={arXiv preprint arXiv:2402.07445},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.07445},
    primaryClass={stat.ML cs.IT cs.LG math.IT math.ST stat.TH}
}