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Online Statistical Inference of Constrained Stochastic Optimization via Random Scaling

Xinchen Du, Wanrong Zhu, Wei Biao Wu, Sen Na

TL;DR

This work tackles online statistical inference for constrained stochastic optimization by developing AI-SSQP, a second-order online solver that uses sketching to inexactly solve quadratic subproblems. It introduces a random-scaling procedure on the averaged AI-SSQP iterates, yielding asymptotically pivotal, matrix-free confidence intervals for the primal-dual solution $(x^*,\lambda^*)$ without explicit covariance estimation. Theoretical results show that the averaged iterates achieve $\sqrt{t}$-consistency with a smaller limiting covariance $\bar{\Xi}^*$ than the last iterate, and the random-scaling test statistic converges to a Brownian-based pivotal distribution, enabling online confidence intervals with $O((d+m)^2)$ per-iteration cost. Numerical experiments on nonlinearly constrained regression demonstrate robust coverage near 95% and competitive interval lengths, with the random-scaling approach outperforming plug-in covariance-based methods under inexact SQP. Overall, the method provides a practical, online, matrix-free framework for inference in constrained stochastic optimization, matching the efficiency of first-order methods while leveraging second-order information for faster convergence.

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 $(\boldsymbol{x}_t, \boldsymbolλ_t)$ 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.

Online Statistical Inference of Constrained Stochastic Optimization via Random Scaling

TL;DR

This work tackles online statistical inference for constrained stochastic optimization by developing AI-SSQP, a second-order online solver that uses sketching to inexactly solve quadratic subproblems. It introduces a random-scaling procedure on the averaged AI-SSQP iterates, yielding asymptotically pivotal, matrix-free confidence intervals for the primal-dual solution without explicit covariance estimation. Theoretical results show that the averaged iterates achieve -consistency with a smaller limiting covariance than the last iterate, and the random-scaling test statistic converges to a Brownian-based pivotal distribution, enabling online confidence intervals with per-iteration cost. Numerical experiments on nonlinearly constrained regression demonstrate robust coverage near 95% and competitive interval lengths, with the random-scaling approach outperforming plug-in covariance-based methods under inexact SQP. Overall, the method provides a practical, online, matrix-free framework for inference in constrained stochastic optimization, matching the efficiency of first-order methods while leveraging second-order information for faster convergence.

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 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.

Paper Structure

This paper contains 24 sections, 18 theorems, 99 equations, 1 figure, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization and notation
  3. Adaptive Inexact Stochastic Sequential Quadratic Programming
  4. Asymptotic Normality of Averaged AI-SSQP
  5. Assumptions and preliminary results
  6. Asymptotic normality
  7. Online Statistical Inference
  8. Numerical Experiments
  9. Experimental setup
  10. Numerical results
  11. Conclusion and Future Work
  12. Preparation Lemmas and Preliminaries
  13. Proofs of Main Theorems
  14. Proof of Theorem \ref{['thm:4']}
  15. Proof of Proposition \ref{['prop:2']}
  16. ...and 9 more sections

Key Result

Theorem 3.4

Consider the AI-SSQP updates in equ:update under Assumptions ass:1, ass:2, ass:3. There exists a threshold $\tau^\star$ such that for any sketching steps $\tau\geq \tau^\star$ and any stepsize control sequences $\{\beta_t, \eta_t = \beta_t+\chi_t\}$ satisfying we have $\|\nabla\mathcal{L}_t\|\rightarrow 0$ and $\|({\boldsymbol{x}}_{t+1}-{\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_{t+1}-{\boldsymbo

Figures (1)

  • Figure 1: The averaged iterate error $\|\bar{{\boldsymbol{x}}}_t - {\boldsymbol{x}}^{\star}\|$ and the last iterate error $\|{\boldsymbol{x}}_t - {\boldsymbol{x}}^{\star}\|$ for constrained linear and logistic regression with varying sketching steps $\tau$. We use $d=20$ and Equi-correlation design with $r=0.2$ as an illustrative example.

Theorems & Definitions (19)

  • Remark 2.1
  • Theorem 3.4: Na2025Statistical, Theorem 4.8
  • Theorem 3.5: Asymptotic normality of averaged AI-SSQP
  • Proposition 3.6
  • Proposition 3.7
  • Theorem 4.1
  • Theorem 4.2
  • Corollary 4.3
  • Lemma A.1
  • Lemma A.2
  • ...and 9 more