Understanding Two-Layer Neural Networks with Smooth Activation Functions

• URL: http://arxiv.org/abs/2507.14177v1
• Date: Fri, 11 Jul 2025 01:55:07 GMT
• Title: Understanding Two-Layer Neural Networks with Smooth Activation Functions
• Abstract summary: This paper aims to understand the training solution, which is obtained by the back-propagation algorithm, of two-layer neural networks.<n>The mechanism contains four main principles: construction of Taylor series expansions, strict partial order of knots, smooth-spline implementation and smooth-continuity restriction.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper aims to understand the training solution, which is obtained by the back-propagation algorithm, of two-layer neural networks whose hidden layer is composed of the units with smooth activation functions, including the usual sigmoid type most commonly used before the advent of ReLUs. The mechanism contains four main principles: construction of Taylor series expansions, strict partial order of knots, smooth-spline implementation and smooth-continuity restriction. The universal approximation for arbitrary input dimensionality is proved and experimental verification is given, through which the mystery of ``black box'' of the solution space is largely revealed. The new proofs employed also enrich approximation theory.

Related papers

• Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations [42.57657519369106]
  This work establishes rigorous first-of-its-kind upper bounds on the generalization error for the method of approximating solutions to Navier-Stokes equations.<n>We trained depth-2 neural networks trained via the unsupervised Physics-Informed Neural Network framework.<n>We provide empirical validation of the suggested activation functions and the corresponding bounds on a PINN setup solving the Taylor-Green vortex benchmark.
  arXiv  Detail & Related papers  (2026-03-24T11:08:34Z)
• Neural Collapse under Gradient Flow on Shallow ReLU Networks for Orthogonally Separable Data [52.737775129027575]
  We show that gradient flow on a two-layer ReLU network for classifying orthogonally separable data provably exhibits Neural Collapse (NC)<n>We reveal the role of the implicit bias of the training dynamics in facilitating the emergence of NC.
  arXiv  Detail & Related papers  (2025-10-24T01:36:19Z)
• Function Forms of Simple ReLU Networks with Random Hidden Weights [1.2289361708127877]
  We investigate the function space dynamics of a two-layer ReLU neural network in the infinite-width limit.<n>We highlight the Fisher information matrix's role in steering learning.<n>This work offers a robust foundation for understanding wide neural networks.
  arXiv  Detail & Related papers  (2025-05-23T13:53:02Z)
• 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)
• Fixing the NTK: From Neural Network Linearizations to Exact Convex Programs [63.768739279562105]
  We show that for a particular choice of mask weights that do not depend on the learning targets, this kernel is equivalent to the NTK of the gated ReLU network on the training data. A consequence of this lack of dependence on the targets is that the NTK cannot perform better than the optimal MKL kernel on the training set.
  arXiv  Detail & Related papers  (2023-09-26T17:42:52Z)
• Globally Optimal Training of Neural Networks with Threshold Activation Functions [63.03759813952481]
  We study weight decay regularized training problems of deep neural networks with threshold activations. We derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network.
  arXiv  Detail & Related papers  (2023-03-06T18:59:13Z)
• Simple initialization and parametrization of sinusoidal networks via their kernel bandwidth [92.25666446274188]
  sinusoidal neural networks with activations have been proposed as an alternative to networks with traditional activation functions. We first propose a simplified version of such sinusoidal neural networks, which allows both for easier practical implementation and simpler theoretical analysis. We then analyze the behavior of these networks from the neural tangent kernel perspective and demonstrate that their kernel approximates a low-pass filter with an adjustable bandwidth.
  arXiv  Detail & Related papers  (2022-11-26T07:41:48Z)
• A Functional-Space Mean-Field Theory of Partially-Trained Three-Layer Neural Networks [49.870593940818715]
  We study the infinite-width limit of a type of three-layer NN model whose first layer is random and fixed. Our theory accommodates different scaling choices of the model, resulting in two regimes of the MF limit that demonstrate distinctive behaviors.
  arXiv  Detail & Related papers  (2022-10-28T17:26:27Z)
• The Sample Complexity of One-Hidden-Layer Neural Networks [57.6421258363243]
  We study a class of scalar-valued one-hidden-layer networks, and inputs bounded in Euclidean norm. We prove that controlling the spectral norm of the hidden layer weight matrix is insufficient to get uniform convergence guarantees. We analyze two important settings where a mere spectral norm control turns out to be sufficient.
  arXiv  Detail & Related papers  (2022-02-13T07:12:02Z)
• 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)
• Linear approximability of two-layer neural networks: A comprehensive analysis based on spectral decay [4.042159113348107]
  We first consider the case of single neuron and show that the linear approximability, quantified by the Kolmogorov width, is controlled by the eigenvalue decay of an associate kernel. We show that similar results also hold for two-layer neural networks.
  arXiv  Detail & Related papers  (2021-08-10T23:30:29Z)
• On Sparsity in Overparametrised Shallow ReLU Networks [42.33056643582297]
  We study the ability of different regularisation strategies to capture solutions requiring only a finite amount of neurons, even on the infinitely wide regime. We establish that both schemes are minimised by functions having only a finite number of neurons, irrespective of the amount of overparametrisation.
  arXiv  Detail & Related papers  (2020-06-18T01:35:26Z)
• How Implicit Regularization of ReLU Neural Networks Characterizes the Learned Function -- Part I: the 1-D Case of Two Layers with Random First Layer [5.969858080492586]
  We consider one dimensional (shallow) ReLU neural networks in which weights are chosen randomly and only the terminal layer is trained. We show that for such networks L2-regularized regression corresponds in function space to regularizing the estimate's second derivative for fairly general loss functionals.
  arXiv  Detail & Related papers  (2019-11-07T13:48:15Z)

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.