  The Straight-Through (ST) estimator is a widely used technique for
back-propagating gradients through discrete random variables. However, this
effective method lacks theoretical justification. In this paper, we show that
ST can be interpreted as the simulation of the projected Wasserstein gradient
flow (pWGF). Based on this understanding, a theoretical foundation is
established to justify the convergence properties of ST. Further, another pWGF
estimator variant is proposed, which exhibits superior performance on
distributions with infinite support,e.g., Poisson distributions. Empirically,
we show that ST and our proposed estimator, while applied to different types of
discrete structures (including both Bernoulli and Poisson latent variables),
exhibit comparable or even better performances relative to other
state-of-the-art methods. Our results uncover the origin of the widespread
adoption of the ST estimator and represent a helpful step towards exploring
alternative gradient estimators for discrete variables.
