On the randomized Euler scheme for SDEs with integral-form drift
Paweł Przybyłowicz, Michał Sobieraj
TL;DR
This work addresses strong approximation of SDEs with integral-form drift $a(x)=\mathbb{E}(H(\xi,x))$ by introducing a randomized Euler scheme that samples drift information with $M$ Monte Carlo points per time step. The authors derive $L^{p}$-error bounds that depend on the time discretization level $n$ and sampling size $M$, with tight rates for several coefficient classes, and show the information cost scales as $\Theta(nM)$. They explore connections to perturbed SGD and demonstrate the method's practical behavior through GPU-based numerical experiments, including optimization scenarios and comparisons with standard optimizers. The results suggest potential improvements via multilevel Monte Carlo to reduce information costs and indicate promising directions for integrating stochastic dynamics with optimization strategies.
Abstract
In this paper, we investigate the problem of strong approximation of the solutions of stochastic differential equations (SDEs) when the drift coefficient is given in integral form. 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 presenting various rates of convergence. Integral-form drift often appears when analyzing stochastic dynamics of optimization procedures in machine learning (ML) problems. Hence, we additionally discuss connections of the defined randomized Euler approximation scheme with the perturbed version of the stochastic gradient descent (SGD) algorithm. Finally, the results of numerical experiments performed using GPU architecture are also reported, including a comparison with other popular optimizers used in ML.