# Malliavin Calculus with Weak Derivatives for Counterfactual Stochastic Optimization

arXiv: https://arxiv.org/abs/2510.00297 · HTML: https://mlsanity.com/paper/2510.00297/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 14 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 1 of 1)

- [Multi-kernel Passive Stochastic Gradient Algorithms and Transfer Learning](https://mlsanity.com/paper/2008.10020/) (2020)

## Built on by (most cited first, 1 of 1)

- [Malliavin Calculus for Counterfactual Gradient Estimation in Adaptive Inverse Reinforcement Learning](https://mlsanity.com/paper/2604.01345/) (2026)

## Abstract (arXiv)

We study counterfactual stochastic optimization of conditional loss functionals under misspecified and noisy gradient information. The difficulty is that when the conditioning event has vanishing or zero probability, naive Monte Carlo estimators are prohibitively inefficient; kernel smoothing, though common, suffers from slow convergence. We propose a two-stage kernel-free methodology. First, we show using Malliavin calculus that the conditional loss functional of a diffusion process admits an exact representation as a Skorohod integral, yielding variance comparable to classical Monte-Carlo variance. Second, we establish that a weak derivative estimate of the conditional loss functional with respect to model parameters can be evaluated with constant variance, in contrast to the widely used score function method whose variance grows linearly in the sample path length. Together, these results yield an efficient framework for counterfactual conditional stochastic gradient algorithms in rare-event regimes.