Operator Splitting with Hamilton-Jacobi-based Proximals

Nicholas Di, Eric Chi, Samy Wu Fung
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:24594-24621, 2026.

Abstract

Operator splitting algorithms are a cornerstone of modern first-order optimization, decomposing complex problems into simpler subproblems solved via proximal operators. However, most functions lack closed-form proximal operators, which has long restricted these methods to a narrow set of problems. Hamilton-Jacobi-based proximal operator (HJ-Prox) is a recent derivative-free Monte Carlo technique based on Hamilton-Jacobi PDE theory, that approximates proximal operators numerically. In this work, we introduce a unified framework for operator splitting via HJ-Prox, which allows for deployment of operator splitting even when functions are not proximable. We prove that replacing exact proximal steps with HJ-Prox in algorithms such as proximal point, proximal gradient descent, Douglas–Rachford splitting, Davis–Yin splitting, and primal–dual hybrid gradient preserves convergence guarantees under mild assumptions. Numerical experiments demonstrate HJ-Prox is competitive and effective on a wide variety of statistical learning tasks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-di26a, title = {Operator Splitting with {H}amilton-Jacobi-based Proximals}, author = {Di, Nicholas and Chi, Eric and Wu Fung, Samy}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {24594--24621}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/di26a/di26a.pdf}, url = {https://proceedings.mlr.press/v306/di26a.html}, abstract = {Operator splitting algorithms are a cornerstone of modern first-order optimization, decomposing complex problems into simpler subproblems solved via proximal operators. However, most functions lack closed-form proximal operators, which has long restricted these methods to a narrow set of problems. Hamilton-Jacobi-based proximal operator (HJ-Prox) is a recent derivative-free Monte Carlo technique based on Hamilton-Jacobi PDE theory, that approximates proximal operators numerically. In this work, we introduce a unified framework for operator splitting via HJ-Prox, which allows for deployment of operator splitting even when functions are not proximable. We prove that replacing exact proximal steps with HJ-Prox in algorithms such as proximal point, proximal gradient descent, Douglas–Rachford splitting, Davis–Yin splitting, and primal–dual hybrid gradient preserves convergence guarantees under mild assumptions. Numerical experiments demonstrate HJ-Prox is competitive and effective on a wide variety of statistical learning tasks.} }
Endnote
%0 Conference Paper %T Operator Splitting with Hamilton-Jacobi-based Proximals %A Nicholas Di %A Eric Chi %A Samy Wu Fung %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-di26a %I PMLR %P 24594--24621 %U https://proceedings.mlr.press/v306/di26a.html %V 306 %X Operator splitting algorithms are a cornerstone of modern first-order optimization, decomposing complex problems into simpler subproblems solved via proximal operators. However, most functions lack closed-form proximal operators, which has long restricted these methods to a narrow set of problems. Hamilton-Jacobi-based proximal operator (HJ-Prox) is a recent derivative-free Monte Carlo technique based on Hamilton-Jacobi PDE theory, that approximates proximal operators numerically. In this work, we introduce a unified framework for operator splitting via HJ-Prox, which allows for deployment of operator splitting even when functions are not proximable. We prove that replacing exact proximal steps with HJ-Prox in algorithms such as proximal point, proximal gradient descent, Douglas–Rachford splitting, Davis–Yin splitting, and primal–dual hybrid gradient preserves convergence guarantees under mild assumptions. Numerical experiments demonstrate HJ-Prox is competitive and effective on a wide variety of statistical learning tasks.
APA
Di, N., Chi, E. & Wu Fung, S.. (2026). Operator Splitting with Hamilton-Jacobi-based Proximals. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:24594-24621 Available from https://proceedings.mlr.press/v306/di26a.html.

Related Material