  In the stochastic gradient descent (SGD) for sequential simulations such as
the neural stochastic differential equations, the Multilevel Monte Carlo (MLMC)
method is known to offer better theoretical computational complexity compared
to the naive Monte Carlo approach. However, in practice, MLMC scales poorly on
massively parallel computing platforms such as modern GPUs, because of its
large parallel complexity which is equivalent to that of the naive Monte Carlo
method. To cope with this issue, we propose the delayed MLMC gradient estimator
that drastically reduces the parallel complexity of MLMC by recycling
previously computed gradient components from earlier steps of SGD. The proposed
estimator provably reduces the average parallel complexity per iteration at the
cost of a slightly worse per-iteration convergence rate. In our numerical
experiments, we use an example of deep hedging to demonstrate the superior
parallel complexity of our method compared to the standard MLMC in SGD.
