Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks

• URL: http://arxiv.org/abs/2607.02003v1
• Date: Thu, 02 Jul 2026 10:39:14 GMT
• Title: Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks
• Abstract summary: We show that a well-posed surrogate control can be used to prove optimal density for a neural network.<n>This allows us to prove optimal density can be obtained by solving a single linear-based system.
• Score: 1.5749416770494706
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics. In this work, we propose a paradigm shift by replacing the discrete training problem of shallow neural networks with a well-posed continuum variational surrogate. We identify a family of $λ$-convex functionals over parameter densities in weighted Sobolev spaces and prove that these variational problems are globally well-posed, stable, and exhibit unexpected almost $C^3$ regularity. Unlike existing Wasserstein-based or Mean-Field approaches, which often face limited regularity and discretization challenges, our formulation provides direct access to elliptic regularity and convex analysis. This allows us to prove that the optimal parameter density can be obtained by solving a single linear system, bypassing iterative optimization entirely. We establish explicit generalization error controls at a rate of $1/α$ relative to the regularization parameter, and prove that finite-width networks of size $N$ achieve the continuum optimum at an $O(1/N)$ rate. This perspective bridges the gap between the Neural Tangent Kernel (NTK) and feature-learning regimes, providing a principled framework for understanding over-parameterization through the lens of variational calculus.

Related papers

• Generalization Bounds of Stochastic Gradient Descent in Homogeneous Neural Networks [29.858071115963472]
  We show that homogeneous networks encompass fully-connected and neural convolutional neural networks with ReReRe activations.<n>This finding is broadly applicable, as homogeneous networks encompass fully-connected and neural networks with ReReRe activations.
  arXiv  Detail & Related papers  (2026-02-26T12:26:32Z)
• Stability and Generalization of Push-Sum Based Decentralized Optimization over Directed Graphs [55.77845440440496]
  Push-based decentralized communication enables optimization over communication networks, where information exchange may be asymmetric.<n>We develop a unified uniform-stability framework for the Gradient Push (SGP) algorithm.<n>A key technical ingredient is an imbalance-aware generalization bound through two quantities.
  arXiv  Detail & Related papers  (2026-02-24T05:32:03Z)
• Smoothness Adaptivity in Constant-Depth Neural Networks: Optimal Rates via Smooth Activations [11.1057367593811]
  We study approximation and statistical properties of neural networks with smooth activations for learning functions in the Sobolev space.<n>We prove that constant-depth networks equipped with smooth activations achieve smoothness adaptivity.<n>We identify activation smoothness as a fundamental mechanism, complementary to depth, for achieving optimal rates over Sobolev function classes.
  arXiv  Detail & Related papers  (2026-02-23T10:38:12Z)
• Error Feedback under $(L_0,L_1)$-Smoothness: Normalization and Momentum [56.37522020675243]
  We provide the first proof of convergence for normalized error feedback algorithms across a wide range of machine learning problems. We show that due to their larger allowable stepsizes, our new normalized error feedback algorithms outperform their non-normalized counterparts on various tasks.
  arXiv  Detail & Related papers  (2024-10-22T10:19:27Z)
• 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)
• A multiobjective continuation method to compute the regularization path of deep neural networks [1.3654846342364308]
  Sparsity is a highly feature in deep neural networks (DNNs) since it ensures numerical efficiency, improves the interpretability of models, and robustness. We present an algorithm that allows for the entire sparse front for the above-mentioned objectives in a very efficient manner for high-dimensional gradients with millions of parameters. We demonstrate that knowledge of the regularization path allows for a well-generalizing network parametrization.
  arXiv  Detail & Related papers  (2023-08-23T10:08:52Z)
• A Robustness Analysis of Blind Source Separation [91.3755431537592]
  Blind source separation (BSS) aims to recover an unobserved signal from its mixture $X=f(S)$ under the condition that the transformation $f$ is invertible but unknown. We present a general framework for analysing such violations and quantifying their impact on the blind recovery of $S$ from $X$. We show that a generic BSS-solution in response to general deviations from its defining structural assumptions can be profitably analysed in the form of explicit continuity guarantees.
  arXiv  Detail & Related papers  (2023-03-17T16:30:51Z)
• Implicit Bias of MSE Gradient Optimization in Underparameterized Neural Networks [0.0]
  We study the dynamics of a neural network in function space when optimizing the mean squared error via gradient flow. We show that the network learns eigenfunctions of an integral operator $T_Kinfty$ determined by the Neural Tangent Kernel (NTK) We conclude that damped deviations offers a simple and unifying perspective of the dynamics when optimizing the squared error.
  arXiv  Detail & Related papers  (2022-01-12T23:28:41Z)
• Neural Control Variates [71.42768823631918]
  We show that a set of neural networks can face the challenge of finding a good approximation of the integrand. We derive a theoretically optimal, variance-minimizing loss function, and propose an alternative, composite loss for stable online training in practice. Specifically, we show that the learned light-field approximation is of sufficient quality for high-order bounces, allowing us to omit the error correction and thereby dramatically reduce the noise at the cost of negligible visible bias.
  arXiv  Detail & Related papers  (2020-06-02T11:17:55Z)
• Loss landscapes and optimization in over-parameterized non-linear systems and neural networks [20.44438519046223]
  We show that wide neural networks satisfy the PL$*$ condition, which explains the (S)GD convergence to a global minimum. We show that wide neural networks satisfy the PL$*$ condition, which explains the (S)GD convergence to a global minimum.
  arXiv  Detail & Related papers  (2020-02-29T17:18:28Z)
• Convex Geometry and Duality of Over-parameterized Neural Networks [70.15611146583068]
  We develop a convex analytic approach to analyze finite width two-layer ReLU networks. We show that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set. In higher dimensions, we show that the training problem can be cast as a finite dimensional convex problem with infinitely many constraints.
  arXiv  Detail & Related papers  (2020-02-25T23:05:33Z)

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.