Geometry-aware Bayesian Optimization in Robotics using Riemannian Matérn Kernels

Bayesian optimization is a data-efficient technique which can be used for\ncontrol parameter tuning, parametric policy adaptation, and structure design in\nrobotics. Many of these problems require optimization of functions defined on\nnon-Euclidean domains like spheres, rotation groups, or spaces of\npositive-definite matrices. To do so, one must place a Gaussian process prior,\nor equivalently define a kernel, on the space of interest. Effective kernels\ntypically reflect the geometry of the spaces they are defined on, but designing\nthem is generally non-trivial. Recent work on the Riemannian Mat\\'ern kernels,\nbased on stochastic partial differential equations and spectral theory of the\nLaplace-Beltrami operator, offers promising avenues towards constructing such\ngeometry-aware kernels. In this paper, we study techniques for implementing\nthese kernels on manifolds of interest in robotics, demonstrate their\nperformance on a set of artificial benchmark functions, and illustrate\ngeometry-aware Bayesian optimization for a variety of robotic applications,\ncovering orientation control, manipulability optimization, and motion planning,\nwhile showing its improved performance.\n