Malliavin Calculus as Stochastic Backpropogation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917281481621504 |
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| author | Oden, Kevin D. |
| author_facet | Oden, Kevin D. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_17013 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Malliavin Calculus as Stochastic Backpropogation Oden, Kevin D. Machine Learning 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. |
| title | Malliavin Calculus as Stochastic Backpropogation |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2602.17013 |