Convergence Rate of the (1+1)-Evolution Strategy with Success-Based Step-Size Adaptation on Convex Quadratic Functions

TL;DR

This paper rigorously analyzes the convergence rate of the (1+1)-ES with success-based step-size adaptation on convex quadratic functions, revealing how the eigenvalues of the Hessian influence optimization speed.

Contribution

It provides the first explicit and rigorous convergence rate analysis of the (1+1)-ES on general convex quadratic functions, considering eigenvalue distribution effects.

Findings

Convergence rate is O(exp(-L / Tr(H))) where L is the smallest eigenvalue.

Generalizes known rates for identity and diagonal Hessians.

Highlights the impact of eigenvalue distribution on optimization speed.

Abstract

The (1+1)-evolution strategy (ES) with success-based step-size adaptation is analyzed on a general convex quadratic function and its monotone transformation, that is, $f(x) = g((x - x^*)^\mathrm{T} H (x - x^*))$, where $g:\mathbb{R}\to\mathbb{R}$ is a strictly increasing function, $H$ is a positive-definite symmetric matrix, and $x^* \in \mathbb{R}^d$ is the optimal solution of $f$. The convergence rate, that is, the decrease rate of the distance from a search point $m_t$ to the optimal solution $x^*$, is proven to be in $O(\exp( - L / \mathrm{Tr}(H) ))$, where $L$ is the smallest eigenvalue of $H$ and $\mathrm{Tr}(H)$ is the trace of $H$. This result generalizes the known rate of $O(\exp(- 1/d ))$ for the case of $H = I_{d}$ ($I_d$ is the identity matrix of dimension $d$) and $O(\exp(- 1/ (d\cdot\xi) ))$ for the case of $H = \mathrm{diag}(\xi \cdot I_{d/2}, I_{d/2})$. To the best of our knowledge, this is the first study in which the convergence rate of the (1+1)-ES is derived explicitly and rigorously on a general convex quadratic function, which depicts the impact of the distribution of the eigenvalues in the Hessian $H$ on the optimization and not only the impact of the condition number of $H$.