Exponential convergence rates for momentum stochastic gradient descent in the overparametrized setting

• URL: http://arxiv.org/abs/2302.03550v2
• Date: Wed, 06 Nov 2024 10:19:32 GMT
• Title: Exponential convergence rates for momentum stochastic gradient descent in the overparametrized setting
• Authors: Benjamin Gess, Sebastian Kassing,
• Abstract summary: We prove bounds on the rate of convergence for the momentum gradient descent scheme (MSGD) We analyze the optimal choice of the friction and show that the MSGD process almost surely converges to a local.
• Score: 0.6445605125467574
• License:
• Abstract: We prove explicit bounds on the exponential rate of convergence for the momentum stochastic gradient descent scheme (MSGD) for arbitrary, fixed hyperparameters (learning rate, friction parameter) and its continuous-in-time counterpart in the context of non-convex optimization. In the small step-size regime and in the case of flat minima or large noise intensities, these bounds prove faster convergence of MSGD compared to plain stochastic gradient descent (SGD). The results are shown for objective functions satisfying a local Polyak-Lojasiewicz inequality and under assumptions on the variance of MSGD that are satisfied in overparametrized settings. Moreover, we analyze the optimal choice of the friction parameter and show that the MSGD process almost surely converges to a local minimum.

Related papers

• Gradient Normalization with(out) Clipping Ensures Convergence of Nonconvex SGD under Heavy-Tailed Noise with Improved Results [60.92029979853314]
  This paper investigates Gradient Normalization without (NSGDC) its gradient reduction variant (NSGDC-VR) We present significant improvements in the theoretical results for both algorithms.
  arXiv  Detail & Related papers  (2024-10-21T22:40:42Z)
• Emergence of heavy tails in homogenized stochastic gradient descent [1.450405446885067]
  Loss by gradient descent (SGD) leads to heavy-tailed network parameters. We analyze a continuous diffusion approximation of SGD, called homogenized gradient descent. We quantify the interplay between optimization parameters and the tail-index.
  arXiv  Detail & Related papers  (2024-02-02T13:06:33Z)
• Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction [49.66486092259376]
  The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of the Langevin dynamics that incorporates a distribution-dependent drift. Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures. We provide a framework to prove a uniform-in-time propagation of chaos for MFLD that takes into account the errors due to finite-particle approximation, time-discretization, and gradient approximation.
  arXiv  Detail & Related papers  (2023-06-12T16:28:11Z)
• Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent [1.2031796234206138]
  It is shown that the inclusion of fluctuations in the limiting SPDE improves the rate of convergence, and retains information about the fluctuations of descent in the continuum limit.
  arXiv  Detail & Related papers  (2022-07-12T17:27:18Z)
• 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)
• Convergence and Stability of the Stochastic Proximal Point Algorithm with Momentum [14.158845925610438]
  We show how a gradient proximal algorithm with momentum (PPA) allows faster convergence to a neighborhood compared to the proximal algorithm (PPA) with better contraction factor.
  arXiv  Detail & Related papers  (2021-11-11T12:17:22Z)
• Minibatch vs Local SGD with Shuffling: Tight Convergence Bounds and Beyond [63.59034509960994]
  We study shuffling-based variants: minibatch and local Random Reshuffling, which draw gradients without replacement. For smooth functions satisfying the Polyak-Lojasiewicz condition, we obtain convergence bounds which show that these shuffling-based variants converge faster than their with-replacement counterparts. We propose an algorithmic modification called synchronized shuffling that leads to convergence rates faster than our lower bounds in near-homogeneous settings.
  arXiv  Detail & Related papers  (2021-10-20T02:25:25Z)
• Differentiable Annealed Importance Sampling and the Perils of Gradient Noise [68.44523807580438]
  Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective. We propose a differentiable algorithm by abandoning Metropolis-Hastings steps, which further unlocks mini-batch computation.
  arXiv  Detail & Related papers  (2021-07-21T17:10:14Z)
• 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)
• 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.