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An analysis of training and generalization errors in shallow and deep networks.

H N Mhaskar1, T Poggio2

  • 1Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, United States of America.

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Summary
This summary is machine-generated.

This study investigates deep networks, finding that specific parameter counts prevent overfitting in regression problems with periodic activation functions. A regularization approach ensures good training and generalization error for compositional functions.

Keywords:
Deep learningGeneralization errorInterpolatory approximation

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Area of Science:

  • Machine Learning
  • Deep Learning Theory
  • Artificial Intelligence

Background:

  • Deep networks often exhibit no overfitting despite over-parameterization.
  • Understanding generalization in deep learning remains a key challenge.
  • Periodic activation functions are common in deep network architectures.

Purpose of the Study:

  • Analyze the absence of overfitting in deep networks with periodic activation functions for regression.
  • Propose alternative loss measures for compositional function approximation.
  • Determine parameter bounds for achieving both zero training and good generalization error.

Main Methods:

  • Analysis of regression problems with periodic activation functions.
  • Measurement of generalization error using maximum and pointwise loss.
  • Estimation of parameter requirements for optimal performance.
  • Proof of regularization problem solution guarantees.

Main Results:

  • Identified parameter ranges that prevent overfitting in specific deep network settings.
  • Demonstrated the inadequacy of minimal expected square loss for compositional functions.
  • Established that regularization yields good training and generalization error.
  • Provided estimates for expected error on test data.

Conclusions:

  • The study offers insights into the generalization capabilities of over-parameterized deep networks.
  • A regularization strategy is validated for achieving robust performance.
  • The findings contribute to a theoretical understanding of deep learning generalization.