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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.07445 |
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| _version_ | 1866911925607071744 |
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| author | Yang, Yuepeng Chen, Antares Orecchia, Lorenzo Ma, Cong |
| author_facet | Yang, Yuepeng Chen, Antares Orecchia, Lorenzo Ma, Cong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07445 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Top-$K$ ranking with a monotone adversary Yang, Yuepeng Chen, Antares Orecchia, Lorenzo Ma, Cong Machine Learning Information Theory Statistics Theory 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. |
| title | Top-$K$ ranking with a monotone adversary |
| topic | Machine Learning Information Theory Statistics Theory |
| url | https://arxiv.org/abs/2402.07445 |