arXiv:2511.03087v1 Announce Type: cross 
Abstract: Generalized linear models (GLMs) are fundamental tools for statistical modeling, with maximum likelihood estimation (MLE) serving as the classical method for parameter inference. While MLE performs well in canonical GLMs, it can become computationally inefficient near the true parameter value. In more general settings with non-canonical or fully general link functions, the resulting optimization landscape is often non-convex, non-smooth, and numerically unstable. To address these challenges, we investigate an alternative estimator based on solving the variational inequality (VI) formulation of the GLM likelihood equations, originally proposed by Juditsky and Nemirovski as an alternative for solving nonlinear least-squares problems. Unlike their focus on algorithmic convergence in monotone settings, we analyze the VI approach from a statistical perspective, comparing it systematically with the MLE. We also extend the theory of VI estimators to a broader class of link functions, including non-monotone cases satisfying a strong Minty condition, and show that it admits weaker smoothness requirements than MLE, enabling faster, more stable, and less locally trapped optimization. Theoretically, we establish both non-asymptotic estimation error bounds and asymptotic normality for the VI estimator, and further provide convergence guarantees for fixed-point and stochastic approximation algorithms. Numerical experiments show that the VI framework preserves the statistical efficiency of MLE while substantially extending its applicability to more challenging GLM settings.

تناقش ورقة حديثة التقدم في أساليب التقدير لنماذج الانحدار الخطي العام (GLMs)، مع تسليط الضوء على قيود تقدير الاحتمالية القصوى (MLE) في سيناريوهات معينة. على الرغم من أن MLE هو نهج قياسي، إلا أنه قد يواجه صعوبات في الكفاءة الحسابية في الإعدادات المعقدة. هذه الأبحاث مهمة لأنها تستكشف تقدير عدم المساواة المتغيرة، والذي يمكن أن يوفر حلولاً أكثر قوة للنمذجة الإحصائية، خاصة في الحالات التي تفشل فيها الطرق التقليدية.

Un artículo reciente discute los avances en los métodos de estimación para modelos lineales generalizados (GLM), destacando las limitaciones de la estimación de máxima verosimilitud (MLE) en ciertos escenarios. Aunque MLE es un enfoque estándar, puede tener problemas de eficiencia computacional en configuraciones complejas. Esta investigación es significativa ya que explora la estimación de desigualdad variacional, que podría proporcionar soluciones más robustas para la modelización estadística, especialmente en casos donde los métodos tradicionales no son efectivos.

Un article récent aborde les avancées dans les méthodes d'estimation pour les modèles linéaires généralisés (GLM), soulignant les limites de l'estimation du maximum de vraisemblance (MLE) dans certains scénarios. Bien que MLE soit une approche standard, elle peut rencontrer des difficultés d'efficacité computationnelle dans des contextes complexes. Cette recherche est importante car elle explore l'estimation par inégalité variationnelle, qui pourrait offrir des solutions plus robustes pour la modélisation statistique, en particulier dans les cas où les méthodes traditionnelles échouent.

A recent paper discusses advancements in the estimation methods for generalized linear models (GLMs), highlighting the limitations of maximum likelihood estimation (MLE) in certain scenarios. While MLE is a standard approach, it can struggle with computational efficiency in complex settings. This research is significant as it explores variational inequality estimation, which could provide more robust solutions for statistical modeling, particularly in cases where traditional methods fall short.

Beyond Maximum Likelihood: Variational Inequality Estimation for Generalized Linear Models

arXiv:2405.09676v2 Announce Type: replace-cross 
Abstract: Classical results in asymptotic statistics show that the Fisher information matrix controls the difficulty of estimating a statistical model from observed data. In this work, we introduce a companion measure of robustness of an estimation problem: the radius of statistical efficiency (RSE) is the size of the smallest perturbation to the problem data that renders the Fisher information matrix singular. We compute RSE up to numerical constants for a variety of testbed problems, including principal component analysis, generalized linear models, phase retrieval, bilinear sensing, and matrix completion. Interestingly, we observe a precise reciprocal relationship between RSE and the intrinsic complexity/sensitivity of the problem instance, paralleling the classical Eckart-Young theorem in numerical analysis. To establish our results, we develop theory for spectral functions of measures that extends well-known results from matrix analysis and eigenvalue optimization$-$a contribution that may be of interest beyond our immediate findings.

دراسة حديثة تقدم مفهوم نصف قطر الكفاءة الإحصائية (RSE)، وهو مقياس جديد يقيس متانة مشاكل التقدير من خلال تحديد أصغر اضطراب يجعل مصفوفة معلومات فيشر مفردة. تشمل هذه الأبحاث مجموعة متنوعة من النماذج الإحصائية، بما في ذلك تحليل المكونات الرئيسية والنماذج الخطية العامة، مما يبرز العلاقة بين RSE وتعقيد هذه النماذج.

Un estudio reciente introduce el radio de eficiencia estadística (RSE), una nueva medida que cuantifica la robustez de los problemas de estimación al determinar la perturbación más pequeña que hace que la matriz de información de Fisher sea singular. Esta investigación abarca varios modelos estadísticos, incluyendo el análisis de componentes principales y modelos lineales generalizados, destacando la relación entre el RSE y la complejidad de estos modelos.

Une étude récente introduit le rayon d'efficacité statistique (RSE), une nouvelle mesure qui quantifie la robustesse des problèmes d'estimation en déterminant la plus petite perturbation qui rend la matrice d'information de Fisher singulière. Cette recherche couvre divers modèles statistiques, y compris l'analyse en composantes principales et les modèles linéaires généralisés, mettant en lumière l'interaction entre le RSE et la complexité de ces modèles.

A recent study introduces the radius of statistical efficiency (RSE), a new measure that quantifies the robustness of estimation problems by determining the smallest perturbation that makes the Fisher information matrix singular. This research spans various statistical models, including principal component analysis and generalized linear models, highlighting the interplay between RSE and the complexity of these models.

Beyond Maximum Likelihood: Variational Inequality Estimation for Generalized Linear Models

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