Fenchel-Young Estimators of Perturbed Utility Models

2026-02-24

AI Summary

This work addresses the numerical instability of traditional maximum likelihood estimation in perturbed utility models, which arises from non-convexity and data sparsity. The authors propose the first unified estimation framework that integrates Fenchel–Young losses with Wasserstein distributionally robust optimization. By leveraging the convex conjugate structure inherent in perturbed utility models, the framework guarantees global convexity and bounded gradients in the objective function. Efficient computation is achieved through safe approximations and finite-dimensional reformulations. Notably, the approach generalizes existing methods—such as L2 regularization and hinge loss—as special cases, offering a theoretically cohesive perspective. Empirical evaluations on synthetic data and the Swissmetro benchmark demonstrate significant performance gains over current state-of-the-art techniques, with robust preference recovery even under severe data scarcity.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

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

Abstract

The Perturbed Utility Model framework offers a powerful generalization of discrete choice analysis, unifying models like Multinomial Logit and Sparsemax through convex optimization. However, standard Maximum Likelihood Estimation (MLE) faces severe theoretical and numerical challenges when applied to this broader class, particularly regarding non-convexity and instability in sparse regimes. To resolve these issues, this paper introduces a unified estimation framework based on the Fenchel-Young loss. By leveraging the intrinsic convex conjugate structure of PUMs, we demonstrate that the Fenchel-Young estimator guarantees global convexity and bounded gradients, providing a mathematically natural alternative to MLE. Addressing the critical challenge of data scarcity, we further extend this framework via Wasserstein Distributionally Robust Optimization. We first derive an exact finite-dimensional reformulation of the infinite-dimensional primal problem, establishing its theoretical convexity. However, recognizing that the resulting worst-case constraints involve computationally intractable inner maximizations, we subsequently construct a tractable safe approximation by exploiting the global Lipschitz continuity of the Fenchel-Young loss. Through this tractable formulation, we uncover a rigorous geometric unification: two canonical regularization techniques, standard L2-regularization and the margin-enforcing Hinge loss, emerge mathematically as specific limiting cases of our distributionally robust estimator. Extensive experiments on synthetic data and the Swissmetro benchmark validate that the proposed framework significantly outperforms traditional methods, recovering stable preferences even under severe data limitations.

Problem

• Perturbed Utility Models
• Maximum Likelihood Estimation
• Non-convexity
• Data Scarcity
• Distributionally Robust Optimization

Innovation

• Fenchel-Young loss
• Perturbed Utility Models
• Distributionally Robust Optimization
• Convex Estimation
• Wasserstein Robustness