Almost Sure Saddle Avoidance of Stochastic Gradient Methods without the Bounded Gradient Assumption

• URL: http://arxiv.org/abs/2302.07862v1
• Date: Wed, 15 Feb 2023 18:53:41 GMT
• Title: Almost Sure Saddle Avoidance of Stochastic Gradient Methods without the Bounded Gradient Assumption
• Authors: Jun Liu, Ye Yuan
• Abstract summary: We prove that various gradient descent methods, including the gradient descent (SGD), heavy-ball (SHB) and Nesterov's accelerated gradient (SNAG) methods, almost surely avoid any strict saddle manifold. This is the first time such results are obtained for SHB and SNAG methods.
• Score: 11.367487348673793
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove that various stochastic gradient descent methods, including the stochastic gradient descent (SGD), stochastic heavy-ball (SHB), and stochastic Nesterov's accelerated gradient (SNAG) methods, almost surely avoid any strict saddle manifold. To the best of our knowledge, this is the first time such results are obtained for SHB and SNAG methods. Moreover, our analysis expands upon previous studies on SGD by removing the need for bounded gradients of the objective function and uniformly bounded noise. Instead, we introduce a more practical local boundedness assumption for the noisy gradient, which is naturally satisfied in empirical risk minimization problems typically seen in training of neural networks.

Related papers

• Gradient Normalization Provably Benefits Nonconvex SGD under Heavy-Tailed Noise [60.92029979853314]
  We investigate the roles of gradient normalization and clipping in ensuring the convergence of Gradient Descent (SGD) under heavy-tailed noise. Our work provides the first theoretical evidence demonstrating the benefits of gradient normalization in SGD under heavy-tailed noise. We introduce an accelerated SGD variant incorporating gradient normalization and clipping, further enhancing convergence rates under heavy-tailed noise.
  arXiv  Detail & Related papers  (2024-10-21T22:40:42Z)
• Limit Theorems for Stochastic Gradient Descent with Infinite Variance [47.87144151929621]
  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.
  arXiv  Detail & Related papers  (2024-10-21T09:39:10Z)
• Flattened one-bit stochastic gradient descent: compressed distributed optimization with controlled variance [55.01966743652196]
  We propose a novel algorithm for distributed gradient descent (SGD) with compressed gradient communication in the parameter-server framework. Our gradient compression technique, named flattened one-bit gradient descent (FO-SGD), relies on two simple algorithmic ideas.
  arXiv  Detail & Related papers  (2024-05-17T21:17:27Z)
• Accelerated gradient methods for nonconvex optimization: Escape trajectories from strict saddle points and convergence to local minima [9.66353475649122]
  This paper considers the problem. of understanding convex behavior of a general general of accelerated gradient methods on. non-aptotic functions. It shows that Nesterov's accelerated method (NAG) with variable momentum avoids strict saddle points almost surely.
  arXiv  Detail & Related papers  (2023-07-13T19:11:07Z)
• 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)
• Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise [64.85879194013407]
  We prove the first high-probability results with logarithmic dependence on the confidence level for methods for solving monotone and structured non-monotone VIPs. Our results match the best-known ones in the light-tails case and are novel for structured non-monotone problems. In addition, we numerically validate that the gradient noise of many practical formulations is heavy-tailed and show that clipping improves the performance of SEG/SGDA.
  arXiv  Detail & Related papers  (2022-06-02T15:21:55Z)
• Computing the Variance of Shuffling Stochastic Gradient Algorithms via Power Spectral Density Analysis [6.497816402045099]
  Two common alternatives to gradient descent (SGD) with theoretical benefits are random reshuffling (SGDRR) and shuffle-once (SGD-SO) We study the stationary variances of SGD, SGDRR and SGD-SO, whose leading terms decrease in this order, and obtain simple approximations.
  arXiv  Detail & Related papers  (2022-06-01T17:08:04Z)
• Stability and Convergence of Stochastic Gradient Clipping: Beyond Lipschitz Continuity and Smoothness [23.22461721824713]
  Gradient clipping is a technique to stabilize the training process for problems prone to the exploding gradient problem. This paper establishes both qualitative and quantitative results of the gradient clipped (sub)gradient method (SGD) for non-smooth convex functions. We also study the convergence of a clipped method with momentum, which includes clipped SGD as a special case.
  arXiv  Detail & Related papers  (2021-02-12T12:41:42Z)
• Path Sample-Analytic Gradient Estimators for Stochastic Binary Networks [78.76880041670904]
  In neural networks with binary activations and or binary weights the training by gradient descent is complicated. We propose a new method for this estimation problem combining sampling and analytic approximation steps. We experimentally show higher accuracy in gradient estimation and demonstrate a more stable and better performing training in deep convolutional models.
  arXiv  Detail & Related papers  (2020-06-04T21:51:21Z)
• Stochastic Optimization with Heavy-Tailed Noise via Accelerated Gradient Clipping [69.9674326582747]
  We propose a new accelerated first-order method called clipped-SSTM for smooth convex optimization with heavy-tailed distributed noise in gradients. We prove new complexity that outperform state-of-the-art results in this case. We derive the first non-trivial high-probability complexity bounds for SGD with clipping without light-tails assumption on the noise.
  arXiv  Detail & Related papers  (2020-05-21T17:05:27Z)
• Stochastic Approximate Gradient Descent via the Langevin Algorithm [11.36635610546803]
  We introduce the approximate gradient descent (SAGD) as an alternative to the gradient descent for cases where unbiased gradients cannot be trivially obtained. We show that SAGD performs well experimentally in popular statistical and machine learning problems such as the expectation-maximization algorithm and the variational autoencoders.
  arXiv  Detail & Related papers  (2020-02-13T14:29:21Z)

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.