Conditioning Diffusions Using Malliavin Calculus

Jakiw Pidstrigach, Elizabeth Louise Baker, Carles Domingo-Enrich, George Deligiannidis, Nikolas Nüsken
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:49292-49315, 2025.

Abstract

In generative modelling and stochastic optimal control, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and centred around a generalisation of the Tweedie score formula to nonlinear stochastic differential equations, that enables the development of methods robust to such singularities. This allows our approach to handle a broad range of applications, like diffusion bridges, or adding conditional controls to an already trained diffusion model. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. As a byproduct, we also introduce a novel score matching objective. Our loss functions are formulated such that they could readily be extended to manifold-valued and infinite dimensional diffusions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-pidstrigach25a, title = {Conditioning Diffusions Using Malliavin Calculus}, author = {Pidstrigach, Jakiw and Baker, Elizabeth Louise and Domingo-Enrich, Carles and Deligiannidis, George and N\"{u}sken, Nikolas}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {49292--49315}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/pidstrigach25a/pidstrigach25a.pdf}, url = {https://proceedings.mlr.press/v267/pidstrigach25a.html}, abstract = {In generative modelling and stochastic optimal control, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and centred around a generalisation of the Tweedie score formula to nonlinear stochastic differential equations, that enables the development of methods robust to such singularities. This allows our approach to handle a broad range of applications, like diffusion bridges, or adding conditional controls to an already trained diffusion model. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. As a byproduct, we also introduce a novel score matching objective. Our loss functions are formulated such that they could readily be extended to manifold-valued and infinite dimensional diffusions.} }
Endnote
%0 Conference Paper %T Conditioning Diffusions Using Malliavin Calculus %A Jakiw Pidstrigach %A Elizabeth Louise Baker %A Carles Domingo-Enrich %A George Deligiannidis %A Nikolas Nüsken %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-pidstrigach25a %I PMLR %P 49292--49315 %U https://proceedings.mlr.press/v267/pidstrigach25a.html %V 267 %X In generative modelling and stochastic optimal control, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and centred around a generalisation of the Tweedie score formula to nonlinear stochastic differential equations, that enables the development of methods robust to such singularities. This allows our approach to handle a broad range of applications, like diffusion bridges, or adding conditional controls to an already trained diffusion model. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. As a byproduct, we also introduce a novel score matching objective. Our loss functions are formulated such that they could readily be extended to manifold-valued and infinite dimensional diffusions.
APA
Pidstrigach, J., Baker, E.L., Domingo-Enrich, C., Deligiannidis, G. & Nüsken, N.. (2025). Conditioning Diffusions Using Malliavin Calculus. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:49292-49315 Available from https://proceedings.mlr.press/v267/pidstrigach25a.html.

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