Generalizing Gaussian Smoothing for Random Search

TL;DR

This paper extends Gaussian smoothing in derivative-free optimization by introducing alternative sampling distributions that reduce gradient estimation error, leading to improved performance in various optimization tasks.

Contribution

It proposes a family of distributions for perturbations that minimize MSE in gradient estimates, outperforming standard Gaussian smoothing.

Findings

New distributions with lower MSE than Gaussian smoothing

Improved optimization performance in regression and reinforcement learning

Competitive or superior results compared to Guided ES and Orthogonal ES

Abstract

Gaussian smoothing (GS) is a derivative-free optimization (DFO) algorithm that estimates the gradient of an objective using perturbations of the current parameters sampled from a standard normal distribution. We generalize it to sampling perturbations from a larger family of distributions. Based on an analysis of DFO for non-convex functions, we propose to choose a distribution for perturbations that minimizes the mean squared error (MSE) of the gradient estimate. We derive three such distributions with provably smaller MSE than Gaussian smoothing. We conduct evaluations of the three sampling distributions on linear regression, reinforcement learning, and DFO benchmarks in order to validate our claims. Our proposal improves on GS with the same computational complexity, and are usually competitive with and often outperform Guided ES and Orthogonal ES, two computationally more expensive algorithms that adapt the covariance matrix of normally distributed perturbations.