Malliavin Calculus as Stochastic Backpropogation

Abstract

We establish a rigorous connection between pathwise (reparameterization) and score-function (Malliavin) gradient estimators by showing that both arise from the Malliavin integration-by-parts identity. Building on this equivalence, we introduce a unified and variance-aware hybrid estimator that adaptively combines pathwise and Malliavin gradients using their empirical covariance structure. The resulting formulation provides a principled understanding of stochastic backpropagation and achieves minimum variance among all unbiased linear combinations, with closed-form finite-sample convergence bounds. We demonstrate 9% variance reduction on VAEs (CIFAR-10) and up to 35% on strongly-coupled synthetic problems. Exploratory policy gradient experiments reveal that non-stationary optimization landscapes present challenges for the hybrid approach, highlighting important directions for future work. Overall, this work positions Malliavin calculus as a conceptually unifying and practically interpretable framework for stochastic gradient estimation, clarifying when hybrid approaches provide tangible benefits and when they face inherent limitations.