Polynomial convergence of iterations of certain random operators in Hilbert space

• URL: http://arxiv.org/abs/2202.02266v1
• Date: Fri, 4 Feb 2022 17:48:29 GMT
• Title: Polynomial convergence of iterations of certain random operators in Hilbert space
• Authors: Soumyadip Ghosh, Yingdong Lu, Tomasz J. Nowicki
• Abstract summary: We study the convergence of random iterative sequence of a family of operators on infinite dimensional spaces inspired bySGD algorithm. We demonstrate that its convergence rate depends on the initial state, while randomness plays a role only in the choice of the best constant factor.
• Score: 2.732936573198251
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the convergence of random iterative sequence of a family of operators on infinite dimensional Hilbert spaces, which are inspired by the Stochastic Gradient Descent (SGD) algorithm in the case of the noiseless regression, as studied in [1]. We demonstrate that its polynomial convergence rate depends on the initial state, while the randomness plays a role only in the choice of the best constant factor and we close the gap between the upper and lower bounds.

Related papers

• Conditioning of Banach Space Valued Gaussian Random Variables: An Approximation Approach Based on Martingales [8.81121308982678]
  We investigate the conditional distributions of two Banach space valued, jointly Gaussian random variables. We show that their means and covariances are determined by a general finite dimensional approximation scheme based upon a martingale approach.
  arXiv  Detail & Related papers  (2024-04-04T13:57:44Z)
• Characterizing Overfitting in Kernel Ridgeless Regression Through the Eigenspectrum [6.749750044497731]
  We prove the phenomena of tempered overfitting and catastrophic overfitting under the sub-Gaussian design assumption. We also identify that the independence of the features plays an important role in guaranteeing tempered overfitting.
  arXiv  Detail & Related papers  (2024-02-02T10:36:53Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance [59.211456992422136]
  We propose algorithms with high-probability convergence results under less restrictive assumptions. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes in optimization.
  arXiv  Detail & Related papers  (2023-02-02T10:37:23Z)
• 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)
• Uniform Function Estimators in Reproducing Kernel Hilbert Spaces [0.0]
  This paper addresses the problem of regression to reconstruct functions, which are observed with superimposed errors at random locations. It is demonstrated that the estimator, which is often derived by employing Gaussian random fields, converges in the mean norm of the kernel reproducing Hilbert space to the conditional expectation.
  arXiv  Detail & Related papers  (2021-08-16T08:13:28Z)
• 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)
• Optimal oracle inequalities for solving projected fixed-point equations [53.31620399640334]
  We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation.
  arXiv  Detail & Related papers  (2020-12-09T20:19:32Z)
• Stochastic optimization with momentum: convergence, fluctuations, and traps avoidance [0.0]
  In this paper, a general optimization procedure is studied, unifying several variants of the gradient descent such as, among others, the heavy ball method, the Nesterov Accelerated Gradient (S-NAG), and the widely used Adam method. The avoidance is studied as a noisy discretization of a non-autonomous ordinary differential equation.
  arXiv  Detail & Related papers  (2020-12-07T19:14:49Z)
• ROOT-SGD: Sharp Nonasymptotics and Near-Optimal Asymptotics in a Single Algorithm [71.13558000599839]
  We study the problem of solving strongly convex and smooth unconstrained optimization problems using first-order algorithms. We devise a novel, referred to as Recursive One-Over-T SGD, based on an easily implementable, averaging of past gradients. We prove that it simultaneously achieves state-of-the-art performance in both a finite-sample, nonasymptotic sense and an sense.
  arXiv  Detail & Related papers  (2020-08-28T14:46:56Z)
• Convergence rates and approximation results for SGD and its continuous-time counterpart [16.70533901524849]
  This paper proposes a thorough theoretical analysis of convex Gradient Descent (SGD) with non-increasing step sizes. First, we show that the SGD can be provably approximated by solutions of inhomogeneous Differential Equation (SDE) using coupling. Recent analyses of deterministic and optimization methods by their continuous counterpart, we study the long-time behavior of the continuous processes at hand and non-asymptotic bounds.
  arXiv  Detail & Related papers  (2020-04-08T18:31:34Z)

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.