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 (1809.03062v3)

Abstract: 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.

• The paper demonstrates that deep learning-based empirical risk minimization effectively overcomes the curse of dimensionality for Black-Scholes PDEs.
• It reformulates high-dimensional PDE approximation as a statistical learning problem by integrating Monte Carlo methods with neural networks.
• Empirical results confirm that both the network size and sample requirements scale polynomially with the problem’s dimension and desired accuracy.

Deep Learning--Based ERM for Black--Scholes PDEs: A Methodological Evaluation

The paper "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" by J. Berner, P. Grohs, and A. Jentzen provides a comprehensive paper on the use of deep learning algorithms to address the curse of dimensionality in solving high-dimensional partial differential equations (PDEs), particularly focusing on the Black–Scholes equation. This document explores the conditions under which empirical risk minimization (ERM) over deep neural network hypothesis classes effectively approximates the solution of $d$-dimensional Kolmogorov equations with affine coefficients in computational finance.

Problem Background and Statement

The researchers consider the challenge of numerically solving PDEs such as the Kolmogorov equation with affine coefficients, whose complexity traditionally scales exponentially with the dimension $d$. This scaling, known as the curse of dimensionality, renders traditional numerical methods inefficient for high-dimensional problems. The paper focuses on deep learning methods to surmount this barrier by demonstrating that, under certain conditions, ERM with deep neural networks can provide a scalable solution in polynomial time relative to both $d$ and the desired approximation accuracy $\varepsilon$.

Methodological Insights

1. Learning Framework and Problem Reformulation: The paper maps the PDE problem onto a statistical learning framework, where the computation of the PDE solution is reformulated as a learning problem. This involves utilizing the Feynman–Kac representation to integrate Monte Carlo methods within a deep learning framework.
2. Neural Network Architecture and Scaling: The authors analyze the capacity of neural networks (with ReLU activation) to approximate linear functions efficiently. They establish that the size of pertinent network parameterization and the number of samples required scale only polynomially with a PDE’s dimension and approximation precision, thus forestalling the curse of dimensionality.
3. Covering Numbers and Generalization Error: The paper employs advanced statistical learning techniques to estimate covering numbers for hypothesis classes formed by neural networks. By determining bounds on the generalization error, they ensure that with high probability, the empirical solution obtained via the network is close to the true solution within the desired precision.
4. Numerical Simulations and Polynomial Bounds: For practical validity, various high-dimensional simulations support the theoretical underpinnings, indicating that the curse of dimensionality is managed through polynomial scaling in network size and sample number. This is thoroughly backed by the probabilistic estimates derived from the research.

Implications and Future Perspectives

The primary implication of this work is its confirmation that deep learning-based ERM can efficiently solve high-dimensional PDEs, specifically in the context of computational finance. The theoretical and empirical evidences consolidate the capability of neural networks to perform robust ERM without succumbing to exponential scaling challenges. Practically, this enhances the applicability of deep learning methods for tackling a broad spectrum of high-dimensional finance problems, such as option pricing in markets characterized by multiple assets.

Theoretically, the research opens pathways for further exploration into applying neural network-based ERM to other classes of PDEs and nonlinear dynamics. Future studies can extend these foundations to address even more intricate equations, such as fully non-linear PDEs, thereby broadening the realms where deep learning can make significant impacts.

In conclusion, the paper’s intricate blend of statistical learning theory, empirical simulations, and rigorous problem reformulation offers a compelling argument and a substantial methodological contribution towards employing deep learning in the numerical approximation of complex high-dimensional systems, paving the way for more innovative approaches in financial mathematics and beyond.