insight - Algorithms and Data Structures - # Online inference for constrained stochastic optimization

Core Concepts

The core message of this paper is to establish the asymptotic normality of the primal-dual iterates generated by an Adaptive Inexact Stochastic Sequential Quadratic Programming (AI-StoSQP) method, which can efficiently solve constrained stochastic optimization problems and perform online statistical inference.

Abstract

The paper considers constrained stochastic nonlinear optimization problems, where the objective function is a stochastic expectation and the constraints are deterministic equalities. To solve these problems, the authors apply the Stochastic Sequential Quadratic Programming (StoSQP) method, which can be viewed as applying a stochastic second-order Newton's method to the Karush-Kuhn-Tucker (KKT) conditions.
To reduce the dominant computational cost of the StoSQP method, the authors propose an Adaptive Inexact StoSQP (AI-StoSQP) scheme that employs an iterative sketching solver to inexactly solve the quadratic program in each iteration. 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 AI-StoSQP method, the authors establish the following key results:
Global almost sure convergence: They show that the KKT residual converges to zero almost surely from any initialization under mild assumptions.
Asymptotic normality: They prove that the rescaled primal-dual sequence 1/√¯αt·(xt-x⋆, λt-λ⋆) converges to a mean-zero Gaussian distribution with a nontrivial covariance matrix depending on the underlying sketching distribution. This result quantifies the uncertainty inherent in the StoSQP iterates, which is crucial for performing online statistical inference.
Covariance estimation: The authors also analyze a plug-in covariance matrix estimator that can be computed in an online fashion to facilitate practical inference.
The authors illustrate the asymptotic normality result on benchmark nonlinear problems in the CUTEst test set and on linearly/nonlinearly constrained regression problems.

Stats

The paper does not provide any specific numerical data or statistics to support the key claims. The results are presented in a theoretical manner.

Quotes

None.

Key Insights Distilled From

by Sen Na,Micha... at **arxiv.org** 04-16-2024

Deeper Inquiries

The proposed AI-StoSQP method can be extended to handle inequality constraints or more general constraint structures beyond equality constraints by incorporating appropriate modifications in the algorithm. One approach is to introduce slack variables to convert inequality constraints into equality constraints. This transformation allows the algorithm to handle both types of constraints simultaneously. Additionally, the augmented Lagrangian method can be adapted to accommodate inequality constraints by introducing penalty terms that penalize violations of the constraints. By incorporating these adjustments, the AI-StoSQP method can effectively handle a broader range of constraint structures in optimization problems.

While the sketching-based approach offers advantages in reducing the computational cost of second-order methods like StoSQP, it also has some limitations compared to other techniques such as quasi-Newton methods or stochastic variance reduction. One limitation is the potential loss of accuracy due to the approximation error introduced by the sketching solver. Inexact solutions to the Newton system may lead to suboptimal convergence rates or compromised solution quality. Additionally, the choice of sketching matrices and the number of sketching steps can impact the performance of the method, requiring careful tuning for optimal results. Furthermore, the sketching-based approach may not be suitable for all types of optimization problems, especially those with specific structural characteristics that may not benefit from sketching techniques.

To strengthen the asymptotic normality results and obtain non-asymptotic finite-sample guarantees on the statistical inference, similar to the work on first-order methods like SGD, several approaches can be considered. One approach is to analyze the concentration properties of the stochastic iterates and derive concentration inequalities that provide bounds on the deviation of the estimates from their true values. By establishing non-asymptotic bounds on the estimation error, researchers can quantify the finite-sample performance of the AI-StoSQP method and provide confidence intervals for the inferred parameters. Additionally, techniques from high-dimensional statistics and random matrix theory can be leveraged to analyze the statistical properties of the method in a finite-sample setting, offering insights into the robustness and accuracy of the inference results.

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