Fast Computational Methods for Regularized Estimating Equations

2026-05-25

AI Summary

This study addresses the computational challenges posed by high-dimensional regularized estimating equations, which often exhibit non-gradient structures, asymmetric Jacobians, over-identification, non-smoothness, non-convexity, or nested optimization, rendering standard penalized methods inefficient. To tackle this, the paper proposes a unified formulation of such problems as fixed-point equations and systematically develops four computational paradigms—minimization-based, Dantzig-type, regularization-based, and fixed-point-based—integrating strategies from penalized optimization, constrained linear programming, iterative root-finding, and proximal fixed-point iterations. This cohesive framework substantially enhances both solvability and algorithmic stability for high-dimensional regularized estimating equations, demonstrating broad applicability to complex settings such as longitudinal data analysis and survival modeling.

Abstract

Estimating equations arise in a wide range of statistical applications, including longitudinal and clustered data analysis, survival analysis, econometrics, and semiparametric inference. In high-dimensional settings, adding sparsity-inducing regularization often leads to computational challenges that are not fully addressed by standard penalized optimization routines. These challenges are closely tied to the structural form of the underlying estimating problem: mainly, the estimating function needs not be the gradient of a scalar objective and may involve asymmetric Jacobians, overidentification, nonsmoothness, nonconvexity, or nested optimization. This article first reviews the application areas of estimating equations, and then the computational methods for regularized estimating equations by organizing them into four broad formulations: minimization-type, Dantzig-type, regularization-type, and fixed-point-type approaches. We discuss the main numerical strategies associated with each formulation, including penalized optimization, constrained linear programming, iterative root-solving, and proximal fixed-point iteration. We also highlight the connection between regularized estimating equations and fixed-point problems, which provides a unified computational perspective for analyzing and solving regularized estimating equations.

Problem

• estimating equations
• regularization
• high-dimensional statistics
• computational challenges
• nonconvexity

Innovation

• regularized estimating equations
• fixed-point formulation
• high-dimensional statistics
• non-convex optimization
• sparsity-inducing regularization