用马里亚维计算统一路径和得分函数梯度，实现更优的随机反向传播。
Malliavin Calculus as Stochastic Backpropogation
- 基于马里亚维积分公式，揭示两种梯度估计方法的内在等价性。
- 提出自适应混合估计算法，在VAE上降低9%方差，强耦合问题降35%。
- 提供可解释框架，指导何时使用混合方法及潜在局限性。
我们通过马里亚维积分-分部恒等式，严格建立了路径（重参数化）与得分函数（马里亚维）梯度估计器之间的联系。基于此等价性，提出一种统一且方差感知的混合估计器，利用经验协方差结构自适应融合路径与马里亚维梯度。该方法为随机反向传播提供了理论基础，并在所有无偏线性组合中实现最小方差，具备闭式有限样本收敛界。在VAE（CIFAR-10）上实现9%方差降低，强耦合合成问题最高达35%。探索性策略梯度实验表明，非平稳优化景观对混合方法构成挑战，揭示未来研究方向。总体而言，本工作将马里亚维微积分确立为随机梯度估计的概念统一与实践可解释框架，明确混合方法的优势与固有局限。
原文摘要 · Abstract (English)
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.
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