Statistical Inference of Constrained Stochastic Optimization via Sketched Sequential Quadratic Programming

TL;DR

This paper develops a method for online statistical inference in constrained stochastic nonlinear optimization using a sketched sequential quadratic programming approach, demonstrating asymptotic normality and practical covariance estimation.

Contribution

It introduces a computationally efficient inexact StoSQP method with sketching, establishing its asymptotic normality for inference in constrained stochastic optimization.

Findings

Convergence to a Gaussian distribution under mild conditions.

Effective covariance estimation via a plug-in method.

Validated on benchmark and regression problems.

Abstract

We consider online statistical inference of constrained stochastic nonlinear optimization problems. We apply the Stochastic Sequential Quadratic Programming (StoSQP) method to solve these problems, which can be regarded as applying second-order Newton's method to the Karush-Kuhn-Tucker (KKT) conditions. In each iteration, the StoSQP method computes the Newton direction by solving a quadratic program, and then selects a proper adaptive stepsize $\bar{\alpha}_t$ to update the primal-dual iterate. To reduce dominant computational cost of the method, we inexactly solve the quadratic program in each iteration by employing an iterative sketching solver. Notably, the approximation error of the sketching solver need not vanish as iterations proceed, meaning that the per-iteration computational cost does not blow up. For the above StoSQP method, we show that under mild assumptions, the rescaled primal-dual sequence $1/\sqrt{\bar{\alpha}_t}\cdot (x_t - x^\star, \lambda_t - \lambda^\star)$ converges to a mean-zero Gaussian distribution with a nontrivial covariance matrix depending on the underlying sketching distribution. To perform inference in practice, we also analyze a plug-in covariance matrix estimator. We illustrate the asymptotic normality result of the method both on benchmark nonlinear problems in CUTEst test set and on linearly/nonlinearly constrained regression problems.