Evolutionary optimization via swarming dynamics on products of spheres and rotation groups

Abstract

We propose novel gradient-free algorithms for optimization problems where the objective functions are defined on products of spheres or rotation groups. Optimization problems of this kind are common in robotics and aeronautics where learning rotations and orientations in space is one of the core tasks. Moreover, in many cases it is required to find several mutually dependent orientations or several coupled rotations, making the optimization problem much more demanding. Our approach is based on recently introduced families of probability distributions, as well as on trainable swarms on spheres and rotation groups. The underlying idea is that models and architectures in robotics and machine learning are to a great extent imposed by geometry of the data. The proposed approach is flexible and can be adapted to setups with sequential (temporal) data. In order to make our methods clearer, a number of illustrative problems are introduced and solved using the proposed methods.