Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations

The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolutionized the area of artificial intelligence, machine learning, and data analysis. In particular, these methods have bee...

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Bibliographic Details
Published in:arXiv.org
Main Authors: Berner, Julius, Grohs, Philipp, Jentzen, Arnulf
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 11.11.2020
Subjects:
Accuracy
Algorithms
Approximation
Artificial intelligence
Artificial neural networks
Computer simulation
Data analysis
Drift
Economic models
Empirical analysis
Error analysis
Hypotheses
Machine learning
Mathematical models
Neural networks
Optimization
Partial differential equations
Regression analysis
Scaling
Statistical analysis
ISSN:2331-8422
Online Access:Get full text
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Summary:The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolutionized the area of artificial intelligence, machine learning, and data analysis. In particular, these methods have been applied to the numerical solution of high-dimensional partial differential equations with great success. Recent simulations indicate that deep learning-based algorithms are capable of overcoming the curse of dimensionality for the numerical solution of Kolmogorov equations, which are widely used in models from engineering, finance, and the natural sciences. The present paper considers under which conditions ERM over a deep neural network hypothesis class approximates the solution of a \(d\)-dimensional Kolmogorov equation with affine drift and diffusion coefficients and typical initial values arising from problems in computational finance up to error \(\varepsilon\). We establish that, with high probability over draws of training samples, such an approximation can be achieved with both the size of the hypothesis class and the number of training samples scaling only polynomially in \(d\) and \(\varepsilon^{-1}\). It can be concluded that ERM over deep neural network hypothesis classes overcomes the curse of dimensionality for the numerical solution of linear Kolmogorov equations with affine coefficients.
Bibliography:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
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ISSN:2331-8422
DOI:10.48550/arxiv.1809.03062