The polar evolution strategy

We show in this paper that the squared norm of an individual subject to Gaussian mutation in an evolution strategy will grow on average linearly with the number of generations. Although we prove this result in the absence of selection, experimental evidence is provided showing that the result also holds in a full fledged evolution strategy applied to the search for projections. This phenomenon implies that regions farther and farther away from the origin are explored as the number of generations increases. This becomes crucial when searching for unit vectors or projections, whose norm should be kept fixed. In order to meet this constraint we propose to change to polar coordinates and use a constrained mutation operator which only mutates angles and keeps the radius constant. Likewise, more complex non-linear equality constraints could be handled by means of general curvilinear coordinates. As an illustrative application, our polar evolution strategy (PES) is applied to a spam filtering problem and a credit card approval problem. The new algorithm is shown to perform as well as other state of the art alternatives.