# Malliavin Calculus for Counterfactual Gradient Estimation in Adaptive Inverse Reinforcement Learning

arXiv: https://arxiv.org/abs/2604.01345 · HTML: https://mlsanity.com/paper/2604.01345/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Inverse reinforcement learning, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 3 of its 20 references that are in the map.
- First posted on arXiv 1 Apr 2026.

## Placed in

- Offline RL and imitation learning from data / [Inverse reinforcement learning](https://mlsanity.com/offlinerl/inverse-reinforcement-learning/)
- Deep reinforcement learning methods / [RL for robot locomotion and manipulation](https://mlsanity.com/deeprl/rl-for-robot-locomotion-and-manipulation/)

## Builds on (most cited first, 3 of 3)

- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Finite-Sample Bounds for Adaptive Inverse Reinforcement Learning using Passive Langevin Dynamics](https://mlsanity.com/paper/2304.09123/) (2023)
- [Malliavin Calculus with Weak Derivatives for Counterfactual Stochastic Optimization](https://mlsanity.com/paper/2510.00297/) (2025)

## Abstract (arXiv)

Inverse reinforcement learning (IRL) recovers the loss function of a forward learner from its observed responses. Adaptive IRL aims to reconstruct the loss function of a forward learner by passively observing its gradients as it performs reinforcement learning (RL). This paper proposes a novel passive Langevin-based algorithm that achieves adaptive IRL. The key difficulty in adaptive IRL is that the required gradients in the passive algorithm are counterfactual, that is, they are conditioned on events of probability zero under the forward learner's trajectory. Therefore, naive Monte Carlo estimators are prohibitively inefficient, and kernel smoothing, though common, suffers from slow convergence. We overcome this by employing Malliavin calculus to efficiently estimate the required counterfactual gradients. We reformulate the counterfactual conditioning as a ratio of unconditioned expectations involving Malliavin quantities, thus recovering standard estimation rates. We derive the necessary Malliavin derivatives and their adjoint Skorohod integral formulations for a general Langevin structure, and provide a concrete algorithmic approach which exploits these for counterfactual gradient estimation.