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Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization

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  • Goda, Takashi
  • Kitade, Wataru

Abstract

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.

Suggested Citation

  • Goda, Takashi & Kitade, Wataru, 2023. "Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 743-763.
    • Handle: RePEc:eee:matcom:v:204:y:2023:i:c:p:743-763
    DOI: 10.1016/j.matcom.2022.09.012
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    References listed on IDEAS

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