Gradient descent provably escapes saddle points in the training of shallow ReLU networks

• URL: http://arxiv.org/abs/2208.02083v2
• Date: Wed, 11 Sep 2024 10:07:56 GMT
• Title: Gradient descent provably escapes saddle points in the training of shallow ReLU networks
• Authors: Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek,
• Abstract summary: We prove a variant of the relevant dynamical systems result, a center-stable manifold theorem, in which we relax some of the regularity requirements. Building on a detailed examination of critical points of the square integral loss function for shallow ReLU and leaky ReLU networks, we show that gradient descents most saddle points.
• Score: 6.458742319938318
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Dynamical systems theory has recently been applied in optimization to prove that gradient descent algorithms bypass so-called strict saddle points of the loss function. However, in many modern machine learning applications, the required regularity conditions are not satisfied. In this paper, we prove a variant of the relevant dynamical systems result, a center-stable manifold theorem, in which we relax some of the regularity requirements. We explore its relevance for various machine learning tasks, with a particular focus on shallow rectified linear unit (ReLU) and leaky ReLU networks with scalar input. Building on a detailed examination of critical points of the square integral loss function for shallow ReLU and leaky ReLU networks relative to an affine target function, we show that gradient descent circumvents most saddle points. Furthermore, we prove convergence to global minima under favourable initialization conditions, quantified by an explicit threshold on the limiting loss.

Related papers

• On the Dynamics Under the Unhinged Loss and Beyond [104.49565602940699]
  We introduce the unhinged loss, a concise loss function, that offers more mathematical opportunities to analyze closed-form dynamics. The unhinged loss allows for considering more practical techniques, such as time-vary learning rates and feature normalization.
  arXiv  Detail & Related papers  (2023-12-13T02:11:07Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• The Implicit Bias of Minima Stability in Multivariate Shallow ReLU Networks [53.95175206863992]
  We study the type of solutions to which gradient descent converges when used to train a single hidden-layer multivariate ReLU network with the quadratic loss. We prove that although shallow ReLU networks are universal approximators, stable shallow networks are not.
  arXiv  Detail & Related papers  (2023-06-30T09:17:39Z)
• Optimal Sets and Solution Paths of ReLU Networks [56.40911684005949]
  We develop an analytical framework to characterize the set of optimal ReLU networks. We establish conditions for the neuralization of ReLU networks to be continuous, and develop sensitivity results for ReLU networks.
  arXiv  Detail & Related papers  (2023-05-31T18:48:16Z)
• Adaptive Self-supervision Algorithms for Physics-informed Neural Networks [59.822151945132525]
  Physics-informed neural networks (PINNs) incorporate physical knowledge from the problem domain as a soft constraint on the loss function. We study the impact of the location of the collocation points on the trainability of these models. We propose a novel adaptive collocation scheme which progressively allocates more collocation points to areas where the model is making higher errors.
  arXiv  Detail & Related papers  (2022-07-08T18:17:06Z)
• Convergence and Implicit Regularization Properties of Gradient Descent for Deep Residual Networks [7.090165638014331]
  We prove linear convergence of gradient descent to a global minimum for the training of deep residual networks with constant layer width and smooth activation function. We show that the trained weights, as a function of the layer index, admits a scaling limit which is H"older continuous as the depth of the network tends to infinity.
  arXiv  Detail & Related papers  (2022-04-14T22:50:28Z)
• Support Vectors and Gradient Dynamics for Implicit Bias in ReLU Networks [45.886537625951256]
  We study gradient flow dynamics in the parameter space when training single-neuron ReLU networks. Specifically, we discover implicit bias in terms of support vectors in ReLU networks, which play a key role in why and how ReLU networks generalize well.
  arXiv  Detail & Related papers  (2022-02-11T08:55:58Z)
• Improved Overparametrization Bounds for Global Convergence of Stochastic Gradient Descent for Shallow Neural Networks [1.14219428942199]
  We study the overparametrization bounds required for the global convergence of gradient descent algorithm for a class of one hidden layer feed-forward neural networks.
  arXiv  Detail & Related papers  (2022-01-28T11:30:06Z)
• Mean-field Analysis of Piecewise Linear Solutions for Wide ReLU Networks [83.58049517083138]
  We consider a two-layer ReLU network trained via gradient descent. We show that SGD is biased towards a simple solution. We also provide empirical evidence that knots at locations distinct from the data points might occur.
  arXiv  Detail & Related papers  (2021-11-03T15:14:20Z)
• Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations [1.8655840060559168]
  We propose a new deep learning-based scheme for solving high dimensional nonlinear backward differential equations (BSDEs) We approximate the unknown solution of a BSDE using a deep neural network and its gradient with automatic differentiation. In order to demonstrate performances of our algorithm, several nonlinear BSDEs including pricing problems in finance are provided.
  arXiv  Detail & Related papers  (2020-10-03T10:18:58Z)

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.