MORE-PLR: multi-output regression employed for partial label ranking

Rank estimation under label noise poses a fundamental challenge, as ordinal annotations often exhibit structured uncertainty rather than simple label corruption. In this paper, we reformulate rank estimation with noisy ordinal labels as a stochastic ordering problem, in which each instance is inherently associated with multiple plausible ranks instead of a single deterministic label. Based on this view, we propose stochastic order learning (SOL), a learning framework that captures ordinal label uncertainty and learns an embedding space through two complementary objectives: a discriminative loss that structures instance--centroid interactions and a stochastic order loss that enforces probabilistic ordering relations between instances. Extensive experiments across diverse datasets demonstrate that SOL enables reliable rank estimation under various types and levels of label noise. The source code is available at https://github.com/cwlee00/SOL.

Calibration, the alignment of predicted probabilities with true outcome frequencies, is essential for reliable decision-making. While extensively studied for classification and regression, calibration has not been formally addressed for probabilistic label ranking, where the goal is to predict a distribution over orderings of a label set. Naively treating rankings as classes ignores their structure and fails to capture important modalities such as pairwise and top-k predictions. We formalize calibration for label ranking and develop a hierarchy of notions covering full rankings, sub-rankings, and top-k rankings. We prove that full-rank calibration implies the others but not conversely, and sub-ranking and top-k calibration are incomparable. Empirically, we find popular label ranking models are often poorly calibrated, with substantial differences between sub-ranking and top-k metrics. Applying our framework to RLHF reward models, we find that calibration correlates strongly but not perfectly with benchmark accuracy, suggesting it captures a meaningful quality dimension beyond top-1 accuracy. These findings motivate future work on understanding the downstream effects of miscalibration and developing methods to correct it.

We study learning a mixture of $k$ Plackett-Luce models from multi-way ranking responses from annotators that may represent heterogeneous underlying preferences. This problem has many applications in AI alignment and preference optimization. Prior work has studied mixtures of Bradley-Terry models from pairwise comparisons. However, estimating a mixture of multi-way ranking models can become theoretically unidentifiable when $k$ exceeds $m/2$, where $m$ is the ranking length. We design an efficient algorithm to address this issue by first augmenting the rankings to a larger size (e.g., generating comparisons from a base model), followed by a gradient-based estimation to reduce inference cost (in the input embedding space). With this procedure in mind, we then fit a mixture of Plackett-Luce (PL) models via an expectation-maximization-style iteration, or MoPLEx in short. We conduct extensive experiments to verify this algorithm. First, we find that the gradient-based approximation estimates true probabilities with less than 5% error on models with up to 34 billion parameters. Second, MoPLEx improves clustering and ranking accuracy by an average of 43.7% and 15.2% over baselines using a single PL model or a mixture of Bradley-Terry models, on UltraFeedback and PERSONA datasets. These results demonstrate the effectiveness of MoPLEx for tackling multi-way rankings following heterogeneous preferences through measuring alignment via gradients.