Statistical Inference of Constrained Stochastic Optimization via Sketched Sequential Quadratic Programming

• URL: http://arxiv.org/abs/2205.13687v4
• Date: Sat, 13 Apr 2024 21:08:29 GMT
• Title: Statistical Inference of Constrained Stochastic Optimization via Sketched Sequential Quadratic Programming
• Authors: Sen Na, Michael W. Mahoney,
• Abstract summary: We consider online statistical inference of constrained nonlinear optimization problems. We apply the Sequential Quadratic Programming (StoSQP) method to solve these problems.
• Score: 53.63469275932989
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Trust-Region Sequential Quadratic Programming for Stochastic Optimization with Random Models [57.52124921268249]
  We propose a Trust Sequential Quadratic Programming method to find both first and second-order stationary points. To converge to first-order stationary points, our method computes a gradient step in each iteration defined by minimizing a approximation of the objective subject. To converge to second-order stationary points, our method additionally computes an eigen step to explore the negative curvature the reduced Hessian matrix.
  arXiv  Detail & Related papers  (2024-09-24T04:39:47Z)
• Fast Unconstrained Optimization via Hessian Averaging and Adaptive Gradient Sampling Methods [0.3222802562733786]
  We consider minimizing finite-sum expectation objective functions via Hessian-averaging based subsampled Newton methods. These methods allow for inexactness and have fixed per-it Hessian approximation costs. We present novel analysis techniques and propose challenges for their practical implementation.
  arXiv  Detail & Related papers  (2024-08-14T03:27:48Z)
• Stochastic Optimization for Non-convex Problem with Inexact Hessian Matrix, Gradient, and Function [99.31457740916815]
  Trust-region (TR) and adaptive regularization using cubics have proven to have some very appealing theoretical properties. We show that TR and ARC methods can simultaneously provide inexact computations of the Hessian, gradient, and function values.
  arXiv  Detail & Related papers  (2023-10-18T10:29:58Z)
• Constrained Optimization via Exact Augmented Lagrangian and Randomized Iterative Sketching [55.28394191394675]
  We develop an adaptive inexact Newton method for equality-constrained nonlinear, nonIBS optimization problems. We demonstrate the superior performance of our method on benchmark nonlinear problems, constrained logistic regression with data from LVM, and a PDE-constrained problem.
  arXiv  Detail & Related papers  (2023-05-28T06:33:37Z)
• Nearly Optimal Linear Convergence of Stochastic Primal-Dual Methods for Linear Programming [5.126924253766052]
  We show that the proposed method exhibits a linear convergence rate for solving sharp instances with a high probability. We also propose an efficient coordinate-based oracle for unconstrained bilinear problems.
  arXiv  Detail & Related papers  (2021-11-10T04:56:38Z)
• Newton-LESS: Sparsification without Trade-offs for the Sketched Newton Update [88.73437209862891]
  In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. We show that the Gaussian sketching matrix can be drastically sparsified, significantly reducing the computational cost of sketching. We prove that Newton-LESS enjoys nearly the same problem-independent local convergence rate as Gaussian embeddings.
  arXiv  Detail & Related papers  (2021-07-15T17:33:05Z)
• A stochastic linearized proximal method of multipliers for convex stochastic optimization with expectation constraints [8.133190610747974]
  We present a computable approximation type algorithm, namely the linearized proximal convex method of multipliers. Some preliminary numerical results demonstrate the performance of the proposed algorithm.
  arXiv  Detail & Related papers  (2021-06-22T07:24:17Z)
• A Momentum-Assisted Single-Timescale Stochastic Approximation Algorithm for Bilevel Optimization [112.59170319105971]
  We propose a new algorithm -- the Momentum- Single-timescale Approximation (MSTSA) -- for tackling problems. MSTSA allows us to control the error in iterations due to inaccurate solution to the lower level subproblem.
  arXiv  Detail & Related papers  (2021-02-15T07:10:33Z)
• An Adaptive Stochastic Sequential Quadratic Programming with Differentiable Exact Augmented Lagrangians [17.9230793188835]
  We consider the problem of solving nonlinear optimization programs with objective and deterministic equality. We propose a sequential quadratic programming (SQP) that uses a differentiable exact augmented Lagrangian as the merit function. The proposed algorithm is the first SQP that allows a line search procedure and the first line search procedure.
  arXiv  Detail & Related papers  (2021-02-10T08:40:55Z)
• Coordinate Methods for Matrix Games [41.821881312775496]
  We develop primal-dual coordinate methods for solving bilinear saddle-point problems of the form $min_x in mathcalX max_yinmathY ytop A x$. Our methods push existing sublinear methods towards their limits in terms of per-iteration complexity and sample complexity.
  arXiv  Detail & Related papers  (2020-09-17T17:55:03Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.