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Uniform Convergence of Deep Neural Networks With Lipschitz Continuous Activation Functions and Variable Widths.

Yuesheng Xu1, Haizhang Zhang2

  • 1Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529 USA.

IEEE Transactions on Information Theory
|July 30, 2025
PubMed
Summary

This study introduces a framework for analyzing deep neural networks (DNNs) with Lipschitz activation functions. It provides conditions for DNNs to uniformly converge as layers increase, including convolutional neural networks.

Keywords:
Lipschitz continuous activation functionsUniform convergenceconvolutional neural networksdeep neural networksvariable widths

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Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Deep Learning Theory

Background:

  • Deep neural networks (DNNs) are powerful machine learning models.
  • Understanding the convergence properties of DNNs, especially as their depth increases, is crucial for theoretical guarantees and practical applications.
  • Lipschitz continuity is a common property of activation functions used in many DNN architectures.

Purpose of the Study:

  • To establish a uniform convergence analysis framework for deep neural networks (DNNs) with Lipschitz continuous activation functions.
  • To provide sufficient conditions on weight matrices, bias vectors, and Lipschitz constants for ensuring uniform convergence of DNNs.
  • To extend the analysis to specific DNN architectures like convolutional neural networks (CNNs).

Main Methods:

  • Development of a theoretical framework for uniform convergence analysis of DNNs.
  • Derivation of conditions on weight matrices and bias vectors for uniform convergence.
  • Analysis of DNNs with fixed, bounded, and unbounded widths.
  • Formulation of conditions on mask sequences for uniform convergence of CNNs.

Main Results:

  • A framework is established to ensure uniform convergence of DNNs as the number of layers tends to infinity.
  • Sufficient conditions for uniform convergence are provided, applicable to various network widths.
  • Specific results are presented for DNNs with fixed, bounded, and unbounded widths.
  • Conditions are derived for the uniform convergence of convolutional neural networks.

Conclusions:

  • The proposed framework guarantees uniform convergence for DNNs with Lipschitz activation functions.
  • The theory accommodates a wide range of commonly used activation functions.
  • The findings contribute to the theoretical understanding of deep learning models, particularly their behavior with increasing depth and width, including CNNs.