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Rectified deep neural networks overcome the curse of dimensionality when approximating solutions of McKean--Vlasov stochastic differential equations

Ariel Neufeld, Tuan Anh Nguyen

TL;DR

This work tackles the challenge of high-dimensional McKean--Vlasov SDEs by proving that rectified deep neural networks can approximate the solutions’ expectations without the curse of dimensionality. The authors build a theoretical framework that combines multilevel Picard approximations with Monte Carlo methods, and then encode these schemes as DNNs using a robust calculus and a perturbation lemma to control error propagation. The main contribution is a polynomial-in-$d$ and polynomial-in-$1/\epsilon$ bound on network size required to achieve accuracy \(\varepsilon\) in the relevant $L^2$ sense, along with explicit representations for MLP and Monte Carlo components as DNNs. This provides a rigorous justification for DNN-based algorithms to solve high-dimensional MV-SDEs and guides future work on forward-backward MV-SDEs with complexity guarantees.

Abstract

In this paper we prove that rectified deep neural networks do not suffer from the curse of dimensionality when approximating McKean--Vlasov SDEs in the sense that the number of parameters in the deep neural networks only grows polynomially in the space dimension $d$ of the SDE and the reciprocal of the accuracy $ε$.

Rectified deep neural networks overcome the curse of dimensionality when approximating solutions of McKean--Vlasov stochastic differential equations

TL;DR

This work tackles the challenge of high-dimensional McKean--Vlasov SDEs by proving that rectified deep neural networks can approximate the solutions’ expectations without the curse of dimensionality. The authors build a theoretical framework that combines multilevel Picard approximations with Monte Carlo methods, and then encode these schemes as DNNs using a robust calculus and a perturbation lemma to control error propagation. The main contribution is a polynomial-in- and polynomial-in- bound on network size required to achieve accuracy in the relevant sense, along with explicit representations for MLP and Monte Carlo components as DNNs. This provides a rigorous justification for DNN-based algorithms to solve high-dimensional MV-SDEs and guides future work on forward-backward MV-SDEs with complexity guarantees.

Abstract

In this paper we prove that rectified deep neural networks do not suffer from the curse of dimensionality when approximating McKean--Vlasov SDEs in the sense that the number of parameters in the deep neural networks only grows polynomially in the space dimension of the SDE and the reciprocal of the accuracy .
Paper Structure (12 sections, 16 theorems, 110 equations)

This paper contains 12 sections, 16 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. A mathematical framework for DNNs
  4. Main result
  5. Outline of the proof and organization of our paper
  6. Perturbation lemma
  7. Deep Neural Networks
  8. Basic DNN calculus
  9. DNN representation of MLP approximations
  10. DNN representation of Monte Carlo approximations
  11. DNN approximations of the expectations
  12. Conclusion

Key Result

Theorem 1.2

Assume m07. Let $T\in(0,\infty)$, $c\in [1,\infty)$, $r\in{\mathbbm{N}}$. For every $d\in {\mathbbm{N}}$, $\varepsilon\in [0,1)$ let $\mu_{d,\varepsilon}\in C({\mathbbm{R}}^d\times{\mathbbm{R}}^d,{\mathbbm{R}}^d)$, $f_{d,\varepsilon}\in C({\mathbbm{R}}^d,{\mathbbm{R}})$. For every $d\in {\mathbbm{N} and Then the following items hold.

Theorems & Definitions (24)

  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1: Existence and uniqueness
  • proof : Proof of \ref{['a01b']}
  • Lemma 2.2: Perturbation lemma
  • proof : Proof of \ref{['a01']}
  • Lemma 3.2: $\odot$ is associative--HJKN2020a
  • Lemma 3.3: $\boxplus$ and associativity--HJKN2020a
  • Lemma 3.4: Triangle inequality--HJKN2020a
  • Lemma 3.5: DNNs for affine transformations--HJKN2020a
  • ...and 14 more