Global Linear Convergence of Evolution Strategies on More than Smooth Strongly Convex Functions

Evolution strategies (ESs) are zeroth-order stochastic black-box optimization heuristics invariant to monotonic transformations of the objective function. They evolve a multivariate normal distribution, from which candidate solutions generated. Among different variants, CMA-ES is nowadays recognized as one state-of-the-art optimizers for difficult problems. Despite ample empirical evidence that ESs with step-size control mechanism converge linearly, theoretical guarantees linear convergence have been established only on limited classes functions. In particular, results convex functions missing, where and also first-order methods often analyzed. this paper, we establish almost sure bound expected hitting time an ES family, namely, $(1+1)_\kappa$-ES, includes (1+1)-ES (generalized) one-fifth success rule abstract covariance matrix adaptation bounded condition number, broad class The analysis holds positively homogeneous quadratically functions, latter particularly transformation strongly Lipschitz continuous gradient. As far authors know, first work proves such