When Do Heavy-Tail Distributions Help?

We examine the evidence for the widespread belief that heavy tail distributions enhance the search for minima on multimodal objective functions. We analyze isotropic and anisotropic heavy-tail Cauchy distributions and investigate the probability to sample a better solution, depending on the step length and the dimensionality of the search space. The probability decreases fast with increasing step length for isotropic Cauchy distributions and moderate search space dimension. The anisotropic Cauchy distribution maintains a large probability for sampling large steps along the coordinate axes, resulting in an exceptionally good performance on the separable multimodal Rastrigin function. In contrast, on a non-separable rotated Rastrigin function or for the isotropic Cauchy distribution the performance difference to a Gaussian search distribution is negligible.