Learning Preference from Observed Rankings

We study preference-based bandits with general reward function classes, where a learner sequentially selects pairs of arms and observes binary preference feedback governed by the Bradley--Terry model. This setting naturally arises in applications such as recommender systems, tournament ranking, and learning from human feedback, where relative preferences are easier to elicit than absolute rewards. The observation model inherits the logistic bandit challenge of handling the problem-dependent constant $κ$, which accounts for the non-linearity of the link function and can grow arbitrarily large. Moreover, prior work has predominantly focused on linear or kernelized reward models, precluding the use of richer function classes. To address these limitations, we consider general reward function classes and introduce the locally sensitive eluder dimension, a novel complexity measure tailored to the logistic structure of preference feedback that yields fine-grained regret guarantees without unfavorable dependence on $κ$. Building on this notion, we propose GINOP (Generic INformative OPtimism), an algorithm that constructs log-loss confidence sets and jointly selects arm pairs to balance optimism and informative exploration. We establish a first-order regret bound that, in contrast with what previous results suggest, demonstrates that learning with preference feedback is as statistically efficient as learning from direct reward observation. Finally, we corroborate our theoretical findings with empirical evaluations against competitive baselines.

Problem definition: This paper studies the problem of estimating consumer preferences from bundle sales data. Product bundling is a widely used pricing strategy in retail markets. To set profitable bundle selection and prices, the seller needs to learn the distribution of consumers' valuations for individual products from the transaction data. When customers purchase bundles or multiple products, classical methods such as discrete choice models cannot be used to estimate consumers' valuations. In this paper, we propose an approach to learn the distribution of consumers' valuations toward the products using bundle sales data. Methodology/results: Our approach is to define a utility model for customer choices and estimate the parameters of a valuation distribution that maximizes the likelihood of observing the transaction data. Our approach reduces this problem to an estimation problem where the samples are censored by polyhedral regions on the valuation space of customers. Using the EM algorithm and Monte Carlo simulation, our approach can recover the distribution of consumers' valuations. We extend the framework to allow for unobserved no-purchases, clustered market segments and to incorporate non-additive bundle utilities with synergy effects. In addition, we provide theoretical results on the identifiability of the probability model and sufficient conditions for local convergence of the EM algorithm. Moreover, the performance of the approach is also demonstrated numerically with synthetic and real datasets. Managerial implications: This study demonstrates the challenge to leverage the transaction data of bundle sales to learn customers' preferences. The proposed algorithm provides a practical guidance for retailers.

Ranking data arise in scientific and machine learning applications, including recommendation systems, information retrieval, voting, marketing, and AI preference ranking from human feedback. Existing statistical work has primarily focused on inference tasks such as preference estimation, rank aggregation, and ranking prediction. However, generating realistic synthetic rankings from an observed population is important for privacy-preserving data sharing, benchmark construction, simulation, and uncertainty quantification. This task is challenging because rankings are high-dimensional combinatorial objects with non-Euclidean dependence structures, while ranking populations often exhibit substantial preference heterogeneity. We propose a framework for population-level generative modeling through a latent preference simplex embedding. It estimates a low-dimensional latent preference simplex through a likelihood-based ranking model, leverages flow matching to learn the population distribution of latent preferences, and generates new rankings through the fitted probabilistic ranking model. We show that ranking generation admits an oracle reduction to latent distribution learning and derive finite-sample generative guarantees that clarify how the number of items, ranking length, and latent dimension affect accuracy. Experiments on synthetic and real datasets demonstrate improved population-level fidelity and provide a statistically interpretable representation of preference heterogeneity.