Convergence Rate of the (1+1)-Evolution Strategy with Success-Based Step-Size Adaptation on Convex Quadratic Functions
This provides a theoretical foundation for understanding how eigenvalue distributions affect optimization in evolutionary strategies, which is incremental but rigorous for the field of optimization algorithms.
The paper analyzes the convergence rate of the (1+1)-evolution strategy with success-based step-size adaptation on convex quadratic functions, proving it to be O(exp(-L/Tr(H))), where L is the smallest eigenvalue and Tr(H) is the trace of the Hessian matrix H, generalizing prior results for specific cases.
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 $O(\exp(- 1/ (d\cdotξ) ))$ for the case of $H = \mathrm{diag}(ξ\cdot I_{d/2}, I_{d/2})$. 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 $H$.