A Hamilton-Jacobi-based Proximal Operator
Abstract
First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit proximal formulas are only known for limited classes of functions. We provide an algorithm, HJ-Prox, for accurately approximating such proximals. This is derived from a collection of relations between proximals, Moreau envelopes, Hamilton-Jacobi (HJ) equations, heat equations, and Monte Carlo sampling. In particular, HJ-Prox smoothly approximates the Moreau envelope and its gradient. The smoothness can be adjusted to act as a denoiser. Our approach applies even when functions are only accessible by (possibly noisy) blackbox samples. We show HJ-Prox is effective numerically via several examples.
Cite
@article{arxiv.2211.12997,
title = {A Hamilton-Jacobi-based Proximal Operator},
author = {Stanley Osher and Howard Heaton and Samy Wu Fung},
journal= {arXiv preprint arXiv:2211.12997},
year = {2023}
}