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[2602.17013] Malliavin Calculus as Stochastic Backpropogation
Summary
This paper establishes a connection between pathwise and score-function gradient estimators in stochastic backpropagation, introducing a hybrid estimator that reduces variance significantly in machine learning applications.
Why It Matters
Understanding the relationship between different stochastic gradient estimation methods is crucial for improving the efficiency and accuracy of machine learning models. This research provides a unified framework that can enhance performance in various applications, particularly in variational autoencoders and policy gradient methods.
Key Takeaways
- Introduces a hybrid estimator combining pathwise and Malliavin gradients.
- Achieves up to 35% variance reduction in synthetic problems and 9% in VAEs.
- Clarifies the conditions under which hybrid approaches are beneficial.
- Highlights challenges in non-stationary optimization landscapes.
- Positions Malliavin calculus as a unifying framework for stochastic gradient estimation.
Computer Science > Machine Learning arXiv:2602.17013 (cs) [Submitted on 2 Nov 2025] Title:Malliavin Calculus as Stochastic Backpropogation Authors:Kevin D. Oden View a PDF of the paper titled Malliavin Calculus as Stochastic Backpropogation, by Kevin D. Oden View PDF HTML (experimental) 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. Subjects: Machine Learning (cs.LG) Cite as: arX...
