Online Statistical Inference for Parameters Estimation with Linear-Equality Constraints

TL;DR

This paper analyzes the asymptotic distribution of projected stochastic gradient descent (PSGD) estimates under linear-equality constraints and introduces an online hypothesis test for these constraints.

Contribution

It provides a theoretical characterization of PSGD's limiting distribution under constraints and proposes an online testing method for linear-equality constraints.

Findings

Theoretical derivation of PSGD's limiting distribution under constraints

Validation of theory through simulations and real data

Development of an online hypothesis testing procedure

Abstract

Stochastic gradient descent (SGD) and projected stochastic gradient descent (PSGD) are scalable algorithms to compute model parameters in unconstrained and constrained optimization problems. In comparison with SGD, PSGD forces its iterative values into the constrained parameter space via projection. From a statistical point of view, this paper studies the limiting distribution of PSGD-based estimate when the true parameters satisfy some linear-equality constraints. Our theoretical findings reveal the role of projection played in the uncertainty of the PSGD-based estimate. As a byproduct, we propose an online hypothesis testing procedure to test the linear-equality constraints. Simulation studies on synthetic data and an application to a real-world dataset confirm our theory.