论文标题
动量改善了Riemannian流形的优化
Momentum Improves Optimization on Riemannian Manifolds
论文作者
论文摘要
我们开发了一种新的Riemannian Descent算法，该算法依赖于动量来改进现有的一阶方法，以进行地理学凸优化。相比之下，仅证明在先前工作中证明的加速收敛率仅显示出强烈的强度符合目标函数。我们进一步将算法扩展到地理上弱的QUASI-CONVEX目标。我们的收敛证明依赖于一个新的估计序列，该顺序说明了收敛速率对歧管曲率的依赖性。我们对在球体上定义的几个优化问题和正确定矩阵的多种优化问题进行经验验证我们的理论结果。
We develop a new Riemannian descent algorithm that relies on momentum to improve over existing first-order methods for geodesically convex optimization. In contrast, accelerated convergence rates proved in prior work have only been shown to hold for geodesically strongly-convex objective functions. We further extend our algorithm to geodesically weakly-quasi-convex objectives. Our proofs of convergence rely on a novel estimate sequence that illustrates the dependency of the convergence rate on the curvature of the manifold. We validate our theoretical results empirically on several optimization problems defined on the sphere and on the manifold of positive definite matrices.
