Limit Theorems for Stochastic Gradient Descent with Infinite Variance

• URL: http://arxiv.org/abs/2410.16340v2
• Date: Fri, 25 Oct 2024 17:32:38 GMT
• Title: Limit Theorems for Stochastic Gradient Descent with Infinite Variance
• Authors: Jose Blanchet, Aleksandar Mijatović, Wenhao Yang,
• Abstract summary: We show that the gradient descent algorithm can be characterized as the stationary distribution of a suitably defined Ornstein-rnstein process driven by an appropriate L'evy process. We also explore the applications of these results in linear regression and logistic regression models.
• Score: 47.87144151929621
• License:
• Abstract: Stochastic gradient descent is a classic algorithm that has gained great popularity especially in the last decades as the most common approach for training models in machine learning. While the algorithm has been well-studied when stochastic gradients are assumed to have a finite variance, there is significantly less research addressing its theoretical properties in the case of infinite variance gradients. In this paper, we establish the asymptotic behavior of stochastic gradient descent in the context of infinite variance stochastic gradients, assuming that the stochastic gradient is regular varying with index $\alpha\in(1,2)$. The closest result in this context was established in 1969 , in the one-dimensional case and assuming that stochastic gradients belong to a more restrictive class of distributions. We extend it to the multidimensional case, covering a broader class of infinite variance distributions. As we show, the asymptotic distribution of the stochastic gradient descent algorithm can be characterized as the stationary distribution of a suitably defined Ornstein-Uhlenbeck process driven by an appropriate stable L\'evy process. Additionally, we explore the applications of these results in linear regression and logistic regression models.

Related papers

• Generalizing Stochastic Smoothing for Differentiation and Gradient Estimation [59.86921150579892]
  We deal with the problem of gradient estimation for differentiable relaxations of algorithms, operators, simulators, and other non-differentiable functions. We develop variance reduction strategies for differentiable sorting and ranking, differentiable shortest-paths on graphs, differentiable rendering for pose estimation, as well as differentiable cryo-ET simulations.
  arXiv  Detail & Related papers  (2024-10-10T17:10:00Z)
• Stochastic Gradient Descent for Gaussian Processes Done Right [86.83678041846971]
  We show that when emphdone right -- by which we mean using specific insights from optimisation and kernel communities -- gradient descent is highly effective. We introduce a emphstochastic dual descent algorithm, explain its design in an intuitive manner and illustrate the design choices. Our method places Gaussian process regression on par with state-of-the-art graph neural networks for molecular binding affinity prediction.
  arXiv  Detail & Related papers  (2023-10-31T16:15:13Z)
• Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent [43.097493761380186]
  gradient algorithms are an efficient method of approximately solving linear systems. We show that gradient descent produces accurate predictions, even in cases where it does not converge quickly to the optimum. Experimentally, gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks.
  arXiv  Detail & Related papers  (2023-06-20T15:07:37Z)
• One-step corrected projected stochastic gradient descent for statistical estimation [49.1574468325115]
  It is based on the projected gradient descent on the log-likelihood function corrected by a single step of the Fisher scoring algorithm. We show theoretically and by simulations that it is an interesting alternative to the usual gradient descent with averaging or the adaptative gradient descent.
  arXiv  Detail & Related papers  (2023-06-09T13:43:07Z)
• A note on diffusion limits for stochastic gradient descent [0.0]
  Much of the theory, that attempts to clarify the role of noise in gradient algorithms, has widely approximated gradient descent by a differential equation with Gaussian noise. We provide a novel theoretical justification for this practice that showcases how the noise arises naturally.
  arXiv  Detail & Related papers  (2022-10-20T13:27:00Z)
• Random-reshuffled SARAH does not need a full gradient computations [61.85897464405715]
  The StochAstic Recursive grAdientritHm (SARAH) algorithm is a variance reduced variant of the Gradient Descent (SGD) algorithm. In this paper, we remove the necessity of a full gradient. The aggregated gradients serve as an estimate of a full gradient in the SARAH algorithm.
  arXiv  Detail & Related papers  (2021-11-26T06:00:44Z)
• Nearest Neighbour Based Estimates of Gradients: Sharp Nonasymptotic Bounds and Applications [0.6445605125467573]
  gradient estimation is of crucial importance in statistics and learning theory. We consider here the classic regression setup, where a real valued square integrable r.v. $Y$ is to be predicted. We prove nonasymptotic bounds improving upon those obtained for alternative estimation methods.
  arXiv  Detail & Related papers  (2020-06-26T15:19:43Z)
• On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration [115.1954841020189]
  We study the inequality and non-asymptotic properties of approximation procedures with Polyak-Ruppert averaging. We prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity.
  arXiv  Detail & Related papers  (2020-04-09T17:54:18Z)
• Generalized Gumbel-Softmax Gradient Estimator for Various Discrete Random Variables [16.643346012854156]
  Esting the gradients of nodes is one of the crucial research questions in the deep generative modeling community. This paper proposes a general version of the Gumbel-Softmax estimator with continuous relaxation.
  arXiv  Detail & Related papers  (2020-03-04T01:13:15Z)

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.