# Lagrangian Inference for Ranking Problems

arXiv: https://arxiv.org/abs/2110.00151 · HTML: https://mlsanity.com/paper/2110.00151/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 131 references that are in the map.
- First posted on arXiv 1 Oct 2021.

## Builds on (most cited first, 23 of 23)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Fairness-Aware Ranking in Search & Recommendation Systems with Application to LinkedIn Talent Search](https://mlsanity.com/paper/1905.01989/) (2019)
- [Fairness in Recommendation Ranking through Pairwise Comparisons](https://mlsanity.com/paper/1903.00780/) (2019)
- [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013)
- [A Deep Look into Neural Ranking Models for Information Retrieval](https://mlsanity.com/paper/1903.06902/) (2019)
- [Simple, Robust and Optimal Ranking from Pairwise Comparisons](https://mlsanity.com/paper/1512.08949/) (2015)
- [Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence](https://mlsanity.com/paper/1505.01462/) (2015)
- [Adversarial Personalized Ranking for Recommendation](https://mlsanity.com/paper/1808.03908/) (2018)
- [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017)
- [Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons](https://mlsanity.com/paper/1504.07218/) (2015)
- [A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models](https://mlsanity.com/paper/1412.8765/) (2014)
- [Hypothesis Testing in High-Dimensional Regression under the Gaussian Random Design Model: Asymptotic Theory](https://mlsanity.com/paper/1301.4240/) (2013)
- [Optimal Full Ranking from Pairwise Comparisons](https://mlsanity.com/paper/2101.08421/) (2021)
- [Learning from Comparisons and Choices](https://mlsanity.com/paper/1704.07228/) (2017)
- [A Nearly Instance Optimal Algorithm for Top-k Ranking under the Multinomial Logit Model](https://mlsanity.com/paper/1707.08238/) (2017)
- [Rank Aggregation via Heterogeneous Thurstone Preference Models](https://mlsanity.com/paper/1912.01211/) (2019)
- [Top-K Ranking from Pairwise Comparisons: When Spectral Ranking is Optimal](https://mlsanity.com/paper/1603.04153/) (2016)
- [High-Dimensional Inference for Cluster-Based Graphical Models](https://mlsanity.com/paper/1806.05139/) (2018)
- [Asymptotically Optimal Sequential Design for Rank Aggregation](https://mlsanity.com/paper/1710.06056/) (2017)
- [Adversarial Top-K Ranking](https://mlsanity.com/paper/1602.04567/) (2016)
- [Bayesian Decision Process for Cost-Efficient Dynamic Ranking via Crowdsourcing](https://mlsanity.com/paper/1612.07222/) (2016)
- [Worst-case vs Average-case Design for Estimation from Fixed Pairwise Comparisons](https://mlsanity.com/paper/1707.06217/) (2017)
- [Individualized Rank Aggregation using Nuclear Norm Regularization](https://mlsanity.com/paper/1410.0860/) (2014)

## Built on by (most cited first, 19 of 19)

- [Ranking Inferences Based on the Top Choice of Multiway Comparisons](https://mlsanity.com/paper/2211.11957/) (2022)
- [Uncertainty Quantification of MLE for Entity Ranking with Covariates](https://mlsanity.com/paper/2212.09961/) (2022)
- [Spectral Ranking Inferences based on General Multiway Comparisons](https://mlsanity.com/paper/2308.02918/) (2023)
- [Sequential Manipulation Against Rank Aggregation: Theory and Algorithm](https://mlsanity.com/paper/2407.01916/) (2024)
- [Contextual Online Uncertainty-Aware Preference Learning for Human Feedback](https://mlsanity.com/paper/2504.19342/) (2025)
- [Ranking from Pairwise Comparisons in General Graphs and Graphs with Locality](https://mlsanity.com/paper/2304.06821/) (2023)
- [Prompt-Dependent Ranking of Large Language Models with Uncertainty Quantification](https://mlsanity.com/paper/2603.03336/) (2026)
- [Reinforcement Learning from Human Feedback: A Statistical Perspective](https://mlsanity.com/paper/2604.02507/) (2026)
- [Statistical inference for pairwise comparison models](https://mlsanity.com/paper/2401.08463/) (2024)
- [Minimax Hypothesis Testing for the Bradley-Terry-Luce Model](https://mlsanity.com/paper/2410.08360/) (2024)
- [Transductive Conformal Inference for Full Ranking](https://mlsanity.com/paper/2501.11384/) (2025)
- [Combinatorial Inference on the Optimal Assortment in Multinomial Logit Models](https://mlsanity.com/paper/2301.12254/) (2023)
- [Rate-Optimal Rank Aggregation with Private Pairwise Rankings](https://mlsanity.com/paper/2402.16792/) (2024)
- [Hypothesis Testing for Generalized Thurstone Models](https://mlsanity.com/paper/2512.02912/) (2025)
- [Entrywise Error Bounds for Spectral Ranking with Semi-Random Adversaries](https://mlsanity.com/paper/2605.23854/) (2026)
- [Ranking a Set of Objects using Heterogeneous Workers: QUITE an Easy Problem](https://mlsanity.com/paper/2310.02016/) (2023)
- [Random pairing MLE for estimation of item parameters in Rasch model](https://mlsanity.com/paper/2406.13989/) (2024)
- [Pairwise Comparisons without Stochastic Transitivity: Model, Theory and Applications](https://mlsanity.com/paper/2501.07437/) (2025)
- [Preference-Based Dynamic Ranking Structure Recognition](https://mlsanity.com/paper/2509.24493/) (2025)

## Abstract (arXiv)

We propose a novel combinatorial inference framework to conduct general uncertainty quantification in ranking problems. We consider the widely adopted Bradley-Terry-Luce (BTL) model, where each item is assigned a positive preference score that determines the Bernoulli distributions of pairwise comparisons' outcomes. Our proposed method aims to infer general ranking properties of the BTL model. The general ranking properties include the "local" properties such as if an item is preferred over another and the "global" properties such as if an item is among the top K-ranked items. We further generalize our inferential framework to multiple testing problems where we control the false discovery rate (FDR), and apply the method to infer the top-K ranked items. We also derive the information-theoretic lower bound to justify the minimax optimality of the proposed method. We conduct extensive numerical studies using both synthetic and real datasets to back up our theory.