Active Bipartite Ranking with Smooth Posterior Distributions

2026-02-27

International Conference on Artificial Intelligence and Statistics

AI Summary

This study addresses bipartite ranking under active learning, extending the setting from discrete conditional distributions to continuous posterior distributions satisfying Hölder smoothness. To this end, the authors propose a novel algorithm, smooth-rank, which adaptively selects samples by minimizing the sup-norm distance between the estimated ranking rule and the optimal ROC curve. The method operates without prior discretization and provides theoretical guarantees within the PAC framework, establishing both upper and lower bounds on the problem-dependent sampling complexity. Both theoretical analysis and empirical experiments demonstrate that smooth-rank significantly outperforms existing approaches in terms of ranking performance and sampling efficiency.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this article, bipartite ranking, a statistical learning problem involved in many applications and widely studied in the passive context, is approached in a much more general \textit{active setting} than the discrete one previously considered in the literature. While the latter assumes that the conditional distribution is piece wise constant, the framework we develop permits in contrast to deal with continuous conditional distributions, provided that they fulfill a H\"older smoothness constraint. We first show that a naive approach based on discretisation at a uniform level, fixed \textit{a priori} and consisting in applying next the active strategy designed for the discrete setting generally fails. Instead, we propose a novel algorithm, referred to as smooth-rank and designed for the continuous setting, which aims to minimise the distance between the ROC curve of the estimated ranking rule and the optimal one w.r.t. the $\sup$ norm. We show that, for a fixed confidence level $\epsilon>0$ and probability $\delta\in (0,1)$, smooth-rank is PAC$(\epsilon,\delta)$. In addition, we provide a problem dependent upper bound on the expected sampling time of smooth-rank and establish a problem dependent lower bound on the expected sampling time of any PAC$(\epsilon,\delta)$ algorithm. Beyond the theoretical analysis carried out, numerical results are presented, providing solid empirical evidence of the performance of the algorithm proposed, which compares favorably with alternative approaches.

Problem

• active learning
• bipartite ranking
• smooth posterior
• Hölder smoothness
• ROC curve

Innovation

• active learning
• bipartite ranking
• smooth posterior
• ROC curve optimization
• PAC guarantee