Deep neural network approximations for Monte Carlo algorithms

Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are also a few results in the literature which prove that DNNs can approximate solutions of certain PDEs without the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially both in the PDE dimension and the reciprocal of the prescribed approximation accuracy. One key argument in most of these results is, first, to use a Monte Carlo approximation scheme which can approximate the solution of the PDE under consideration at a fixed space-time point without the curse of dimensionality and, thereafter, to prove that DNNs are flexible enough to mimic the behaviour of the used approximation scheme. Having this in mind, one could aim for a general abstract result which shows under suitable assumptions that if a certain function can be approximated by any kind of (Monte Carlo) approximation scheme without the curse of dimensionality, then this function can also be approximated with DNNs without the curse of dimensionality. It is a key contribution of this article to make a first step towards this direction. In particular, the main result of this paper, essentially, shows that if a function can be approximated by means of some suitable discrete approximation scheme without the curse of dimensionality and if there exist DNNs which satisfy certain regularity properties and which approximate this discrete approximation scheme without the curse of dimensionality, then the function itself can also be approximated with DNNs without the curse of dimensionality. As an application of this result we establish that solutions of suitable Kolmogorov PDEs can be approximated with DNNs without the curse of dimensionality.