High dimensions and heavy tails for natural evolution strategies

The family of natural evolution strategies (NES) offers a principled approach to real-valued evolutionary optimization. NES follows the natural gradient of the expected fitness on the parameters of its search distribution. While general in its formulation, previous research has focused on multivariate Gaussian search distributions. Here we exhibit problem classes for which other search distributions are more appropriate, and then derive corresponding NES-variants. First, for separable distributions we obtain SNES, whose complexity is only O(d) instead of O(d3). We apply SNES to problems of previously unattainable dimensionality, recovering lowest-energy structures on the Lennard-Jones atom clusters, and obtaining state-of-the-art results on neuro-evolution benchmarks. Second, we develop a new, equivalent formulation based on invariances. This allows for generalizing NES to heavy-tailed distributions, even those with undefined variance, which aids in overcoming deceptive local optima.