On Asymptotic Linear Convergence of Projected Gradient Descent for Constrained Least Squares

• URL: http://arxiv.org/abs/2112.11760v1
• Date: Wed, 22 Dec 2021 09:49:51 GMT
• Title: On Asymptotic Linear Convergence of Projected Gradient Descent for Constrained Least Squares
• Authors: Trung Vu and Raviv Raich
• Abstract summary: This manuscript presents a unified framework for the analysis of projected gradient descent in the context of constrained least squares. We present a recipe for the convergence analysis of PGD and demonstrate it via a beginning-to-end application of the recipe on four fundamental problems.
• Score: 22.851500417035947
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Many recent problems in signal processing and machine learning such as compressed sensing, image restoration, matrix/tensor recovery, and non-negative matrix factorization can be cast as constrained optimization. Projected gradient descent is a simple yet efficient method for solving such constrained optimization problems. Local convergence analysis furthers our understanding of its asymptotic behavior near the solution, offering sharper bounds on the convergence rate compared to global convergence analysis. However, local guarantees often appear scattered in problem-specific areas of machine learning and signal processing. This manuscript presents a unified framework for the local convergence analysis of projected gradient descent in the context of constrained least squares. The proposed analysis offers insights into pivotal local convergence properties such as the condition of linear convergence, the region of convergence, the exact asymptotic rate of convergence, and the bound on the number of iterations needed to reach a certain level of accuracy. To demonstrate the applicability of the proposed approach, we present a recipe for the convergence analysis of PGD and demonstrate it via a beginning-to-end application of the recipe on four fundamental problems, namely, linearly constrained least squares, sparse recovery, least squares with the unit norm constraint, and matrix completion.

Related papers

• Low-Rank Extragradient Methods for Scalable Semidefinite Optimization [0.0]
  We focus on high-dimensional and plausible settings in which the problem admits a low-rank solution. We provide several theoretical results proving that, under these circumstances, the well-known Extragradient method converges to a solution of the constrained optimization problem.
  arXiv  Detail & Related papers  (2024-02-14T10:48:00Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterization [48.65416821017865]
  This chapter introduces a new algorithmic approach, dubbed scaled gradient (ScaledGD) It converges linearly at a constant rate independent of the condition number of the low-rank object. It maintains the low periteration cost of gradient descent for a variety of tasks.
  arXiv  Detail & Related papers  (2023-10-09T21:16:57Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• On the Convergence of Stochastic Extragradient for Bilinear Games with Restarted Iteration Averaging [96.13485146617322]
  We present an analysis of the ExtraGradient (SEG) method with constant step size, and present variations of the method that yield favorable convergence. We prove that when augmented with averaging, SEG provably converges to the Nash equilibrium, and such a rate is provably accelerated by incorporating a scheduled restarting procedure.
  arXiv  Detail & Related papers  (2021-06-30T17:51:36Z)
• Exact Linear Convergence Rate Analysis for Low-Rank Symmetric Matrix Completion via Gradient Descent [22.851500417035947]
  Factorization-based gradient descent is a scalable and efficient algorithm for solving the factorrank matrix completion. We show that gradient descent enjoys fast convergence to estimate a solution of the global nature problem.
  arXiv  Detail & Related papers  (2021-02-04T03:41:54Z)
• Asymptotic convergence rate of Dropout on shallow linear neural networks [0.0]
  We analyze the convergence on objective functions induced by Dropout and Dropconnect, when applying them to shallow linear Neural Networks. We obtain a local convergence proof of the gradient flow and a bound on the rate that depends on the data, the rate probability, and the width of the NN.
  arXiv  Detail & Related papers  (2020-12-01T19:02:37Z)
• Iterative Pre-Conditioning for Expediting the Gradient-Descent Method: The Distributed Linear Least-Squares Problem [0.966840768820136]
  This paper considers the multi-agent linear least-squares problem in a server-agent network. The goal for the agents is to compute a linear model that optimally fits the collective data points held by all the agents, without sharing their individual local data points. We propose an iterative pre-conditioning technique that mitigates the deleterious effect of the conditioning of data points on the rate of convergence of the gradient-descent method.
  arXiv  Detail & Related papers  (2020-08-06T20:01:18Z)
• Conditional gradient methods for stochastically constrained convex minimization [54.53786593679331]
  We propose two novel conditional gradient-based methods for solving structured convex optimization problems. The most important feature of our framework is that only a subset of the constraints is processed at each iteration. Our algorithms rely on variance reduction and smoothing used in conjunction with conditional gradient steps, and are accompanied by rigorous convergence guarantees.
  arXiv  Detail & Related papers  (2020-07-07T21:26:35Z)
• Optimization of Graph Total Variation via Active-Set-based Combinatorial Reconditioning [48.42916680063503]
  We propose a novel adaptive preconditioning strategy for proximal algorithms on this problem class. We show that nested-forest decomposition of the inactive edges yields a guaranteed local linear convergence rate. Our results suggest that local convergence analysis can serve as a guideline for selecting variable metrics in proximal algorithms.
  arXiv  Detail & Related papers  (2020-02-27T16:33:09Z)

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.