Active Bipartite Ranking with Smooth Posterior Distributions
This work addresses the problem of active bipartite ranking for applications requiring continuous data, offering a novel theoretical framework and algorithm with empirical validation, though it is incremental in extending prior discrete methods.
The paper tackles bipartite ranking in an active learning setting by extending it from discrete to continuous conditional distributions with Hölder smoothness, proposing the smooth-rank algorithm that is PAC(ε,δ) and provides theoretical bounds on sampling time, with numerical results showing favorable performance compared to alternatives.
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 Hölder 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 $ε>0$ and probability $δ\in (0,1)$, smooth-rank is PAC$(ε,δ)$. 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$(ε,δ)$ 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.