Large Scale Black-Box Optimization by Limited-Memory Matrix Adaptation

Abstract

The covariance matrix adaptation evolution strategy (CMA-ES) is a popular method to deal with nonconvex and/or stochastic optimization problems when gradient information is not available. Being based on the CMA-ES, the recently proposed matrix adaptation evolution strategy (MA-ES) establishes the rather surprising result that the covariance matrix and all associated operations (e.g., potentially unstable eigen decomposition) can be replaced by an iteratively updated transformation matrix without any loss of performance. In order to further simplify MA-ES and reduce its $\boldsymbol {\mathcal {O}}({n}^{ { {2}}})$ time and storage complexity to $\boldsymbol {\mathcal {O}}({{mn}})$ with $\boldsymbol {m}~{\boldsymbol \ll }~{n}$ such as $\boldsymbol {m}~{\boldsymbol \in }~\boldsymbol {\mathcal {O}}{(1)}$ or ${m} {\in } \boldsymbol {\mathcal {O}}({\log {({n})}})$ , we present the limited-memory MA-ES for efficient zeroth order large-scale optimization. The algorithm demonstrates state-of-the-art performance on a set of established large-scale benchmarks.