On the randomized Euler scheme for SDEs with integral-form drift

Abstract: In this paper, we investigate the problem of strong approximation of the solution of SDEs in the case when the drift coefficient is given in the integral form. Such drift often appears when analyzing stochastic dynamics of optimization procedures in machine learning problems. We discuss connections of the defined randomized Euler approximation scheme with the perturbed version of the stochastic gradient descent (SGD) algorithm. We investigate its upper error bounds, in terms of the discretization parameter n and the size M of the random sample drawn at each step of the algorithm, in different subclasses of coefficients of the underlying SDE. Finally, the results of numerical experiments performed by using GPU architecture are also reported.