论文标题
汉密尔顿 - 雅各比的近端运营商
A Hamilton-Jacobi-based Proximal Operator
论文作者
论文摘要
一阶优化算法如今已广泛使用。这些算法中的两个标准构建块是近端算子（近端）和梯度。尽管可以针对各种函数计算梯度，但显式近端公式仅在有限的功能类别中闻名。我们提供了一种算法HJ-Prox，用于准确近似此类接近。这是从近端，莫罗信封，汉密尔顿 - 雅各布（HJ）方程，热方程和蒙特卡洛采样之间的一系列关系中得出的。特别是，HJ-Prox平滑地近似于Moreau Invelope及其梯度。可以调整平滑度以充当Dinoiser。即使仅通过（可能是嘈杂的）黑框样本才能访问功能，我们的方法也适用。我们显示HJ-Prox通过几个示例在数值上是有效的。
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.
