Beyond Maximum Likelihood: Variational Inequality Estimation for Generalized Linear Models

원문 초록

Generalized linear models (GLMs) are fundamental tools for statistical modeling, with maximum likelihood estimation (MLE) serving as the classical approach for parameter inference. While MLE performs well for canonical GLMs, it can become computationally challenging in more general settings with non-canonical, non-smooth, or saturating link functions, where the resulting optimization landscape may be ill-conditioned, non-convex, or non-differentiable. In this paper, we study an alternative estimation framework based on variational inequalities (VIs), which formulates GLM estimation through an operator-based equilibrium condition rather than likelihood minimization. We analyze the VI estimator from a statistical perspective and establish finite-sample error bounds and asymptotic normality, together with convergence guarantees for fixed-point and stochastic approximation algorithms. The framework accommodates a broad class of link functions, including non-canonical, non-smooth, and non-monotone cases satisfying a strong Minty-type condition, and extends naturally to generalized additive models (GAMs) via basis expansion. Numerical experiments demonstrate that under a fixed iteration budget, the VI approach achieves competitive estimation accuracy and faster, more stable convergence than MLE, particularly in GLMs and GAMs with non-canonical or non-smooth link functions.