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.

翻译：本文研究了当漂移系数以积分形式给出时，随机微分方程解的强逼近问题。此类漂移项常在分析机器学习问题中优化过程的随机动力学时出现。我们讨论了所定义的随机化欧拉逼近格式与随机梯度下降算法的扰动版本之间的联系。我们针对底层随机微分方程系数的不同子类，以离散化参数n和算法每步抽取的随机样本大小M为度量，研究了该格式的误差上界。最后，报告了基于GPU架构进行的数值实验结果。