We study stochastic gradient descent for solving conditional stochastic optimization problems, in which an objective to be minimized is given by a parametric nested expectation with an outer expectation taken with respect to one random variable and an inner conditional expectation with respect to the other random variable. The gradient of such a parametric nested expectation is again expressed as a nested expectation, which makes it hard for the standard nested Monte Carlo estimator to be unbiased. In this paper, we show under some conditions that a multilevel Monte Carlo gradient estimator is unbiased and has finite variance and finite expected computational cost, so that the standard theory from stochastic optimization for a parametric (non-nested) expectation directly applies. We also discuss a special case for which yet another unbiased gradient estimator with finite variance and cost can be constructed.

翻译：我们研究随机梯度梯度下降,以解决有条件的随机随机优化问题,通过对一个随机变量的外延预期和对另一个随机变量的内端预期,将目标降到最低,对一个随机变量的外延预期,对另一个随机变量的内限预期,将这种对准梯度的预期再次表示为嵌入预期,这使得标准嵌入的蒙特卡洛测深仪很难做到不偏不倚。在本文中,我们在某些条件下显示,多层次的蒙特卡洛梯度估计仪是公正的,具有有限的差异和有限的预期计算成本,因此直接适用从对准(非免责的)预期的外延优化得出的标准理论。我们还讨论了一个特殊案例,可以为此再构建另一个具有有限差异和成本的无偏差梯度估计仪。