[Paper Review] Ranking Median Regression: Learning to Order through Local Consensus

Stéphan Clémençon, Anna Korba+1 more|arXiv (Cornell University)|Oct 31, 2017

Game Theory and Voting Systems22 references3 citations

TL;DR

This paper introduces ranking median regression, a method for predicting user preferences as permutations using input features, by leveraging local Kemeny medians to approximate conditional ranking medians. It establishes theoretical learning rates and proposes efficient k-NN and tree-based algorithms (CRIT) that outperform baselines on synthetic and real-world GSS data, achieving lower empirical Kendall distance risks.

ABSTRACT

This article is devoted to the problem of predicting the value taken by a random permutation $Σ$, describing the preferences of an individual over a set of numbered items $\{1,\; \ldots,\; n\}$ say, based on the observation of an input/explanatory r.v. $X$ e.g. characteristics of the individual), when error is measured by the Kendall $τ$ distance. In the probabilistic formulation of the 'Learning to Order' problem we propose, which extends the framework for statistical Kemeny ranking aggregation developped in \citet{CKS17}, this boils down to recovering conditional Kemeny medians of $Σ$ given $X$ from i.i.d. training examples $(X_1, Σ_1),\; \ldots,\; (X_N, Σ_N)$. For this reason, this statistical learning problem is referred to as extit{ranking median regression} here. Our contribution is twofold. We first propose a probabilistic theory of ranking median regression: the set of optimal elements is characterized, the performance of empirical risk minimizers is investigated in this context and situations where fast learning rates can be achieved are also exhibited. Next we introduce the concept of local consensus/median, in order to derive efficient methods for ranking median regression. The major advantage of this local learning approach lies in its close connection with the widely studied Kemeny aggregation problem. From an algorithmic perspective, this permits to build predictive rules for ranking median regression by implementing efficient techniques for (approximate) Kemeny median computations at a local level in a tractable manner. In particular, versions of $k$-nearest neighbor and tree-based methods, tailored to ranking median regression, are investigated. Accuracy of piecewise constant ranking median regression rules is studied under a specific smoothness assumption for $Σ$'s conditional distribution given $X$.

Motivation & Objective

• To formalize the problem of predicting user preference permutations from input features as a statistical learning task.
• To extend statistical Kemeny ranking aggregation to conditional settings with input-dependent preferences.
• To develop computationally efficient, interpretable predictive models based on local consensus for ranking prediction.
• To establish theoretical learning rates and approximation bounds for piecewise constant ranking rules.
• To validate the approach empirically on synthetic data and real-world GSS survey data.

Proposed method

• Formulates ranking median regression as empirical risk minimization over the Kendall τ distance to find conditional Kemeny medians given input X.
• Introduces local Kemeny consensus: approximating conditional medians by computing medians within local regions of the input space.
• Adapts k-nearest neighbors to ranking regression by computing local Kemeny medians over neighborhoods of training points.
• Proposes the CRIT algorithm (Consensus Ranking Ing Tree), a tree-based method that recursively partitions input space to minimize local ranking risk.
• Uses a node impurity measure based on local Kemeny median risk to guide tree splitting, ensuring low variability in predicted rankings within regions.
• Employs Plackett-Luce models and K-means clustering as baseline comparisons in experiments.

Experimental results

Research questions

• RQ1 What is the theoretical foundation for predicting permutations as conditional Kemeny medians given input features?
• RQ2 Can local consensus approximations of Kemeny medians achieve fast learning rates in ranking median regression?
• RQ3 How do k-NN and tree-based methods based on local Kemeny medians compare to baseline models in terms of predictive accuracy?
• RQ4 What structural patterns in input features are revealed by the CRIT algorithm’s splits on real-world preference data?
• RQ5 To what extent can piecewise constant predictive rules approximate optimal ranking median functions under smoothness assumptions?

Key findings

• The CRIT algorithm achieved an average empirical risk of 2.763 on the GSS dataset, significantly outperforming the baseline model.
• The k-NN method achieved an average empirical risk of 3.235 on the same dataset, demonstrating effectiveness on real-world preference data.
• On synthetic data with φ ≥ 2, both k-NN and CRIT successfully recovered the true piecewise constant partition of the input space.
• The CRIT algorithm’s splits revealed meaningful groupings: occupation at the first level, race at the second, and education level at lower levels, consistent with prior analyses.
• Theoretical analysis shows that fast learning rates are achievable when the conditional pairwise probabilities P(Σ(i) < Σ(j) | X = x) are Lipschitz continuous.
• The local consensus approach enables efficient computation via existing Kemeny median solvers, making the method both tractable and interpretable.