Analysis of a Natural Gradient Algorithm on Monotonic Convex-Quadratic-Composite Functions

TL;DR

This paper analyzes a natural gradient-based algorithm for optimizing composite functions, demonstrating its covariance adaptation to the inverse Hessian and comparing its performance to CMA-ES through simulations.

Contribution

It introduces a novel natural gradient variant that adapts covariance matrices to the inverse Hessian, with theoretical convergence analysis and empirical comparisons.

Findings

Covariance matrix becomes proportional to the inverse Hessian.

The algorithm converges at a quantifiable speed.

Stochastic approximation closely matches the deterministic algorithm.

Abstract

In this paper we investigate the convergence properties of a variant of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). Our study is based on the recent theoretical foundation that the pure rank-mu update CMA-ES performs the natural gradient descent on the parameter space of Gaussian distributions. We derive a novel variant of the natural gradient method where the parameters of the Gaussian distribution are updated along the natural gradient to improve a newly defined function on the parameter space. We study this algorithm on composites of a monotone function with a convex quadratic function. We prove that our algorithm adapts the covariance matrix so that it becomes proportional to the inverse of the Hessian of the original objective function. We also show the speed of covariance matrix adaptation and the speed of convergence of the parameters. We introduce a stochastic algorithm that approximates the natural gradient with finite samples and present some simulated results to evaluate how precisely the stochastic algorithm approximates the deterministic, ideal one under finite samples and to see how similarly our algorithm and the CMA-ES perform.