## AbstractBased on the theory of non-negative supermartingales, convergence results are proven for adaptive $(1,\lambda)-ES$ (i.e. with Gaussian mutations), and geometrical convergence rates are derived. In the $d$-dimensional case ($d > 1$), the algorithm studied here uses a different step-size update in each direction. However, the critical value for the step-size, and the resulting convergence rate do not depend on the dimension. Those results are discussed with respect to previous work. Finally, rigourous numerical investigations on some 1-dimensional functions validate the theoretical results.
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