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<p>Nonparametric regression with random design is considered. The <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> error with integration with respect to the design measure is used as error criterion. Over-parametrized deep neural network estimates are defined with logistic activation function where all parameters are learned by stochastic gradient descent. It is shown that the estimates achieve a nearly optimal rate of convergence in case that the regression function is (<i>p</i>,&#xa0;<i>C</i>)–smooth. In case that the regression function satisfies a projection pursuit model or more generally a hierarchical composition model the estimate achieves a rate of convergence which does not depend on the input dimension.</p>

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Rate of convergence of over-parametrized deep neural network regression estimates learned by stochastic gradient descent

  • 期刊论文

  • 发表日期：

    2026年1月3日

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  • Michael Kohler,
  • Adam Krzyżak

摘要

Nonparametric regression with random design is considered. The \(L_2\) L 2 error with integration with respect to the design measure is used as error criterion. Over-parametrized deep neural network estimates are defined with logistic activation function where all parameters are learned by stochastic gradient descent. It is shown that the estimates achieve a nearly optimal rate of convergence in case that the regression function is (p, C)–smooth. In case that the regression function satisfies a projection pursuit model or more generally a hierarchical composition model the estimate achieves a rate of convergence which does not depend on the input dimension.