Online Statistical Inference of Constrained Stochastic Optimization via Random Scaling

TL;DR

This paper introduces a novel online inference method for constrained stochastic optimization using random scaling, enabling real-time decision-making with valid confidence intervals and no matrix inversions.

Contribution

It develops a new online inference procedure called random scaling based on SSQP iterates, providing asymptotically valid confidence intervals without matrix computations.

Findings

The method produces valid confidence intervals in constrained stochastic optimization.

Numerical experiments show superior performance over existing inference procedures.

The approach is matrix-free and suitable for real-time applications.

Abstract

Constrained stochastic nonlinear optimization problems have attracted significant attention for their ability to model complex real-world scenarios in physics, economics, and biology. As datasets continue to grow, online inference methods have become crucial for enabling real-time decision-making without the need to store historical data. In this work, we develop an online inference procedure for constrained stochastic optimization by leveraging a method called Sketched Stochastic Sequential Quadratic Programming (SSQP). As a direct generalization of sketched Newton methods, SSQP approximates the objective with a quadratic model and the constraints with a linear model at each step, then applies a sketching solver to inexactly solve the resulting subproblem. Building on this design, we propose a new online inference procedure called random scaling. In particular, we construct a test statistic based on SSQP iterates whose limiting distribution is free of any unknown parameters. Compared to existing online inference procedures, our approach offers two key advantages: (i) it enables the construction of asymptotically valid confidence intervals; and (ii) it is matrix-free, i.e. the computation involves only primal-dual SSQP iterates $(\boldsymbol{x}_t, \boldsymbol{\lambda}_t)$ without requiring any matrix inversions. We validate our theory through numerical experiments on nonlinearly constrained regression problems and demonstrate the superior performance of our random scaling method over existing inference procedures.