運動最適化におけるガウス・ニュートン法の本質的重要性について
On the Fundamental Importance of Gauss-Newton in Motion Optimization
運動最適化においてガウス・ニュートン近似が真のヘッセ行列に時間離散化を細かくすると急速に収束することを証明し、計算効率の良い運動エネルギー再定式化を提案した論文。
著者: Nathan Ratliff, Marc Toussaint, Jeannette Bohg, Stefan Schaal
分類: cs.RO
原文アブストラクト
Hessian information speeds convergence substantially in motion optimization. The better the Hessian approximation the better the convergence. But how good is a given approximation theoretically? How much are we losing? This paper addresses that question and proves that for a particularly popular and empirically strong approximation known as the Gauss-Newton approximation, we actually lose very little--for a large class of highly expressive objective terms, the true Hessian actually limits to the Gauss-Newton Hessian quickly as the trajectory's time discretization becomes small. This result both motivates it's use and offers insight into computationally efficient design. For instance, traditional representations of kinetic energy exploit the generalized inertia matrix whose derivatives are usually difficult to compute. We introduce here a novel reformulation of rigid body kinetic energy designed explicitly for fast and accurate curvature calculation. Our theorem proves that the Gauss-Newton Hessian under this formulation efficiently captures the kinetic energy curvature, but requires only as much computation as a single evaluation of the traditional representation. Additionally, we introduce a technique that exploits these ideas implicitly using Cholesky decompositions for some cases when similar objective terms reformulations exist but may be difficult to find. Our experiments validate these findings and demonstrate their use on a real-world motion optimization system for high-dof motion generation.
