Convergence Rate of the (1+1)-ES on Locally Strongly Convex and Lipschitz Smooth Functions
研究了(1+1)-ES在局部强凸且梯度Lipschitz连续函数上的线性收敛速度，给出了上下界，且算法无需目标函数的Lipschitz常数等先验信息。
Evolution strategy (ES) is one of the promising classes of algorithms for black-box continuous optimization. Despite its broad successes in applications, theoretical analysis on the speed of its convergence is limited on convex quadratic functions and their monotonic transformation. In this study, an upper bound and a lower bound of the rate of linear convergence of the (1+1)-ES on locally <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> -strongly convex functions with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$U$ </tex-math></inline-formula> -Lipschitz continuous gradient are derived as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\exp (-\Omega _{d\to \infty }({L}/({d\cdot U})))$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\exp (-1/d)$ </tex-math></inline-formula> , respectively. Notably, any prior knowledge on the mathematical properties of the objective function, such as the Lipschitz constant, is not given to the algorithm, whereas the existing analyses of derivative-free optimization algorithms require it.