# HJ-Prox

tl;dr - We give a formula for estimating proximal operators from (possibly noisy) observations of objective function values.

### 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 importance 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.

### Graphics

Video Overview of HJ-Prox
Example of HJ-Prox algorithm using an increasing number of samples to form an estimate (green dot) of the proximal (red dot). Samples
$\mathsf{y^i}$
are drawn from a Gaussian distribution centered at
$\mathsf{x}$
, and a convex combination of these samples is used to get the estimate, with the weight of each sample a function of the objective value
$\mathsf{f(y^i)}$
.

### Slides

HJ-Prox Slides
HJ_Prox_Slides.pdf
1MB
PDF