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 known for only 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 accessible only by (possibly noisy) black box samples. We show that HJ-Prox is effective numerically via several examples.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Mar 29, 2023
- Source ID
- 10.1073/pnas.2220469120
Entities
People
- Howard Heaton
- Samy Wu Fung
- Stanley Osher
Organizations
- Air Force Office of Scientific Research
- Colorado School of Mines
- National Science Foundation
- Office of Naval Research
- University of California