Fenchel-Young Estimators of Perturbed Utility Models

Abstract

The Perturbed Utility Model (PUM) framework provides a generalization of discrete choice analysis, unifying models like Multinomial Logit (MNL) and Sparsemax through convex optimization. However, standard Maximum Likelihood Estimation (MLE) encounters theoretical and computational limitations when applied to this broader class, particularly regarding non-convexity and instability in sparse regimes. To address these issues, this paper introduces a unified estimation framework for PUMs based on the Fenchel-Young loss. By leveraging the intrinsic convex conjugate structure of the choice probabilities, we demonstrate that the Fenchel-Young estimator guarantees global convexity, providing a stable alternative to MLE that accommodates both dense and sparse choice kernels. Furthermore, we establish the framework's asymptotic consistency and normality under standard regularity conditions. Leveraging the tractability of the Fenchel-Young estimator, we further develop a Parametric Basis Estimation (PBE) procedure that estimate utility parameters jointly with a tree-structured perturbation function within a pre-specified basis family. PBE employs a bi-level optimization architecture that parameterizes the unknown perturbation as a learnable convex combination of basis functions. For any fixed perturbation structure, the inner Fenchel--Young estimation problem is globally convex in the utility parameters, yielding a well-defined solution mapping that can be differentiated under regularity conditions. Empirical validation on the Swissmetro dataset demonstrates that the proposed framework improves predictive performance, as measured by the Brier score and Brier Skill Score, compared to the standard MNL baseline.