Flexible Regularized Estimating Equations: Some New Perspectives

TL;DR

This paper explores new theoretical connections between regularized estimating equations, fixed-point problems, and variational inequalities, providing insights that could advance computational and theoretical research in this area.

Contribution

It establishes novel equivalences between regularized estimating equations, fixed-point problems, and variational inequalities, extending these relations to general cases.

Findings

Regularized estimating equations are equivalent to fixed-point problems via proximal operators.

They are also equivalent to generalized variational inequalities.

These equivalences apply broadly to various estimating equations and penalty functions.

Abstract

In this note, we make some observations about the equivalences between regularized estimating equations, fixed-point problems and variational inequalities. A summary of our findings is given below: (a) A regularized estimating equation is equivalent to a fixed-point problem, specified by the proximal operator of the corresponding penalty; (b) A regularized estimating equation is equivalent to a generalized variational inequality; (c) Both equivalences extend to any estimating equations and any penalty functions. To our knowledge, these observations have never been presented in the literature before. We hope our new findings can lead to further research in both computational and theoretical aspects of regularized estimating equations.