Stochastic Gradient Descent outperforms Gradient Descent in recovering a high-dimensional signal in a glassy energy landscape

• URL: http://arxiv.org/abs/2309.04788v2
• Date: Mon, 18 Dec 2023 09:04:57 GMT
• Title: Stochastic Gradient Descent outperforms Gradient Descent in recovering a high-dimensional signal in a glassy energy landscape
• Abstract summary: Gradient Descent (SGD) is an out-of-equilibrium algorithm used extensively to train artificial neural networks. We show that SGD largely outperforms GD for sufficiently small batch sizes.
• Score: 4.73194777046253
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Stochastic Gradient Descent (SGD) is an out-of-equilibrium algorithm used extensively to train artificial neural networks. However very little is known on to what extent SGD is crucial for to the success of this technology and, in particular, how much it is effective in optimizing high-dimensional non-convex cost functions as compared to other optimization algorithms such as Gradient Descent (GD). In this work we leverage dynamical mean field theory to benchmark its performances in the high-dimensional limit. To do that, we consider the problem of recovering a hidden high-dimensional non-linearly encrypted signal, a prototype high-dimensional non-convex hard optimization problem. We compare the performances of SGD to GD and we show that SGD largely outperforms GD for sufficiently small batch sizes. In particular, a power law fit of the relaxation time of these algorithms shows that the recovery threshold for SGD with small batch size is smaller than the corresponding one of GD.

Related papers

• Factor Augmented High-Dimensional SGD [4.542148087324063]
  gradient descent (SGD) is a fundamental optimization algorithm widely used in modern machine learning.<n>We propose Factor-Augmented SGD (FSGD), a new optimization method that leverages latent factor representations in high-dimensional learning tasks.
  arXiv  Detail & Related papers  (2026-05-19T03:10:33Z)
• GS^2: Graph-based Spatial Distribution Optimization for Compact 3D Gaussian Splatting [50.2045409672451]
  3D Gaussian Splatting (3DGS) has demonstrated breakthrough performance in novel view synthesis and real-time rendering.<n>It is constrained by the high memory cost due to a huge number of Gaussian points.<n>We propose graph-based spatial distribution optimization for compact 3DGS.
  arXiv  Detail & Related papers  (2026-04-02T10:41:51Z)
• Non-Singularity of the Gradient Descent map for Neural Networks with Piecewise Analytic Activations [53.348574336527854]
  We investigate the neural network map as a function on the space of weights and biases.<n>We prove, for the first time, the non-singularity of the gradient descent (GD) map on the loss landscape of realistic neural network architectures.
  arXiv  Detail & Related papers  (2025-10-28T14:34:33Z)
• Steepest Descent Density Control for Compact 3D Gaussian Splatting [72.54055499344052]
  3D Gaussian Splatting (3DGS) has emerged as a powerful real-time, high-resolution novel view.<n>We propose a theoretical framework that demystifies and improves density control in 3DGS.<n>We introduce SteepGS, incorporating steepest density control, a principled strategy that minimizes loss while maintaining a compact point cloud.
  arXiv  Detail & Related papers  (2025-05-08T18:41:38Z)
• More Optimal Fractional-Order Stochastic Gradient Descent for Non-Convex Optimization Problems [2.5971517743176915]
  We propose 2SED Fractional-Order Gradient Descent (2FOSGD), which integrates the Two-Scale Effective Dimension (2SED) with FOSGD.<n>By tracking sensitivity and effective dimensionality, 2SEDFOSGD dynamically modulates the exponent to mitigate sluggish oscillations and hasten convergence.
  arXiv  Detail & Related papers  (2025-05-05T19:27:36Z)
• Effective Dimension Aware Fractional-Order Stochastic Gradient Descent for Convex Optimization Problems [2.5971517743176915]
  We propose 2SED Fractional-Order Gradient Descent (2SEDFOSGD) to adapt the fractional exponent in a data-driven manner.<n>Theoretically, this approach preserves the advantages of fractional memory without the sluggish or unstable behavior observed in na"ive fractional SGD.
  arXiv  Detail & Related papers  (2025-03-17T22:57:37Z)
• TeZO: Empowering the Low-Rankness on the Temporal Dimension in the Zeroth-Order Optimization for Fine-tuning LLMs [58.19080159470868]
  We propose a novel low-rank ZO estimator, TeZO, which captures the low-rankness across both the model and temporal dimension. Specifically, we represent ZO perturbations along the temporal dimension as a 3D tensor and employ Canonical Polyadic Decomposition (CPD) to extract each low-rank 2D matrix.
  arXiv  Detail & Related papers  (2025-01-31T11:34:03Z)
• The Optimality of (Accelerated) SGD for High-Dimensional Quadratic Optimization [4.7256945641654164]
  gradient descent (SGD) is a widely used algorithm in machine learning, particularly for neural network training. Recent studies on SGD for canonical quadratic optimization or linear regression show it attains well generalization under suitable high-dimensional settings. This paper investigates SGD with two essential components in practice: exponentially decaying step size schedule and momentum.
  arXiv  Detail & Related papers  (2024-09-15T14:20:03Z)
• Massive Dimensions Reduction and Hybridization with Meta-heuristics in Deep Learning [0.24578723416255746]
  Histogram-based Differential Evolution (HBDE) hybridizes gradient-based and gradient-free algorithms to optimize parameters. HBDE outperforms baseline gradient-based and parent gradient-free DE algorithms evaluated on CIFAR-10 and CIFAR-100 datasets.
  arXiv  Detail & Related papers  (2024-08-13T20:28:20Z)
• LD-GAN: Low-Dimensional Generative Adversarial Network for Spectral Image Generation with Variance Regularization [72.4394510913927]
  Deep learning methods are state-of-the-art for spectral image (SI) computational tasks. GANs enable diverse augmentation by learning and sampling from the data distribution. GAN-based SI generation is challenging since the high-dimensionality nature of this kind of data hinders the convergence of the GAN training yielding to suboptimal generation. We propose a statistical regularization to control the low-dimensional representation variance for the autoencoder training and to achieve high diversity of samples generated with the GAN.
  arXiv  Detail & Related papers  (2023-04-29T00:25:02Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Genetically Modified Wolf Optimization with Stochastic Gradient Descent for Optimising Deep Neural Networks [0.0]
  This research aims to analyze an alternative approach to optimizing neural network (NN) weights, with the use of population-based metaheuristic algorithms. A hybrid between Grey Wolf (GWO) and Genetic Modified Algorithms (GA) is explored, in conjunction with Gradient Descent (SGD) This algorithm allows for a combination between exploitation and exploration, whilst also tackling the issue of high-dimensionality.
  arXiv  Detail & Related papers  (2023-01-21T13:22:09Z)
• DR-DSGD: A Distributionally Robust Decentralized Learning Algorithm over Graphs [54.08445874064361]
  We propose to solve a regularized distributionally robust learning problem in the decentralized setting. By adding a Kullback-Liebler regularization function to the robust min-max optimization problem, the learning problem can be reduced to a modified robust problem. We show that our proposed algorithm can improve the worst distribution test accuracy by up to $10%$.
  arXiv  Detail & Related papers  (2022-08-29T18:01:42Z)
• Adaptive Step-Size Methods for Compressed SGD [15.32764898836189]
  Compressed decentralized Gradient Descent (SGD) algorithms have been recently proposed to address the communication bottleneck in distributed and decentralized networks. We introduce a scaling step in which we use to establish order- convergence rates for compressed datasets. We present experimental results on this for real-world datasets.
  arXiv  Detail & Related papers  (2022-07-20T17:20:58Z)
• Low-Precision Stochastic Gradient Langevin Dynamics [70.69923368584588]
  We provide the first study of low-precision Gradient Langevin Dynamics, showing that its costs can be significantly reduced without sacrificing performance. We develop a new quantization function for SGLD that preserves the variance in each update step. We demonstrate that low-precision SGLD achieves comparable performance to full-precision SGLD with only 8 bits on a variety of deep learning tasks.
  arXiv  Detail & Related papers  (2022-06-20T17:25:41Z)
• Distributed stochastic optimization with large delays [59.95552973784946]
  One of the most widely used methods for solving large-scale optimization problems is distributed asynchronous gradient descent (DASGD) We show that DASGD converges to a global optimal implementation model under same delay assumptions.
  arXiv  Detail & Related papers  (2021-07-06T21:59:49Z)
• Direction Matters: On the Implicit Bias of Stochastic Gradient Descent with Moderate Learning Rate [105.62979485062756]
  This paper attempts to characterize the particular regularization effect of SGD in the moderate learning rate regime. We show that SGD converges along the large eigenvalue directions of the data matrix, while GD goes after the small eigenvalue directions.
  arXiv  Detail & Related papers  (2020-11-04T21:07:52Z)
• Stochastic Runge-Kutta methods and adaptive SGD-G2 stochastic gradient descent [0.0]
  We introduce a second order Runge Kutta method and show that it yields a consistent procedure for the minimization of the loss function. In addition it can be coupled in an adaptive framework, with a Gradient Descent (SGD) to adjust to automatically adjust the learning rate of the SGD. The adaptive SGD, called SGD-G2, is successfully tested on standard datasets.
  arXiv  Detail & Related papers  (2020-02-20T15:45:53Z)

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.