Bayesian optimization is a data-efficient technique which can be used for control parameter tuning, parametric policy adaptation, and structure design in robotics. Many of these problems require optimization of functions defined on non-Euclidean domains like spheres, rotation groups, or spaces of positive-definite matrices. To do so, one must place a Gaussian process prior, or equivalently define a kernel, on the space of interest. Effective kernels typically reflect the geometry of the spaces they are defined on, but designing them is generally non-trivial. Recent work on the Riemannian Mat\'ern kernels, based on stochastic partial differential equations and spectral theory of the Laplace-Beltrami operator, offers promising avenues towards constructing such geometry-aware kernels. In this paper, we study techniques for implementing these kernels on manifolds of interest in robotics, demonstrate their performance on a set of artificial benchmark functions, and illustrate geometry-aware Bayesian optimization for a variety of robotic applications, covering orientation control, manipulability optimization, and motion planning, while showing its improved performance.

翻译：贝叶斯优化是一种数据高效的技术，可用于机器人学中的控制参数调优、参数化策略自适应以及结构设计。许多此类问题需要对定义在非欧几里得域（如球面、旋转群或正定矩阵空间）上的函数进行优化。为此，须在目标空间上设定高斯过程先验，即等价于定义一个核函数。有效的核函数通常能反映其定义空间的几何结构，但设计这类核函数通常具有挑战性。基于随机偏微分方程和拉普拉斯-贝尔特拉米算子谱理论的最新研究——即黎曼马特恩核，为构建此类几何感知核提供了有前景的途径。本文研究了在机器人学相关流形上实现这些核的技术，通过一组人工基准函数验证其性能，并针对多种机器人应用（涵盖姿态控制、可操作性优化和运动规划）展示了基于几何感知的贝叶斯优化，同时证明了其更优的性能表现。