Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

TL;DR

This paper demonstrates that deep neural networks can efficiently approximate value functions in complex, high-dimensional stochastic control problems involving stiff systems, breaking the curse of dimensionality.

Contribution

It establishes polynomial complexity bounds for DNN representations of value functions in controlled stiff SDEs, including those from PDE and SPDE approximations.

Findings

DNNs can approximate value functions with polynomial complexity in high dimensions.

The approach applies to systems with regime switching and Lévy noise.

DNNs overcome the curse of dimensionality in stochastic control of PDEs and SPDEs.

Abstract

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This implies that DNNs can break the curse of dimensionality in numerical approximations and optimal control of PDEs and SPDEs. The main ingredient of our proof is to construct a suitable discrete-time system to effectively approximate the evolution of the underlying stochastic dynamics. Similar ideas can also be applied to obtain expression rates of DNNs for value functions induced by stiff systems with regime switching coefficients and driven by general L\'{e}vy noise.