The study of accelerated gradient methods in Riemannian optimization has recently witnessed notable progress. However, in contrast with the Euclidean setting, a systematic understanding of acceleration is still lacking in the Riemannian setting. We revisit the \emph{Accelerated Hybrid Proximal Extragradient} (A-HPE) method of \citet{monteiro2013accelerated}, a powerful framework for obtaining accelerated Euclidean methods. Subsequently, we propose a Riemannian version of A-HPE. The basis of our analysis of Riemannian A-HPE is a set of insights into Euclidean A-HPE, which we combine with a careful control of distortion caused by Riemannian geometry. We describe a number of Riemannian accelerated gradient methods as concrete instances of our framework.

翻译：最近,里曼尼优化中加速梯度方法的研究取得了显著进展,然而,与欧几里德环境相比,里曼环境仍缺乏对加速度的系统理解。我们重新审视了里曼环境的加速混合加速性超强度方法(A-HPE),这是获得加速欧几里德方法的强大框架。随后,我们提议了里曼环境的A-HPE版本。我们分析里曼环境的A-HPE的基础就是一套对欧几里德环境的深入了解,我们结合了对里曼地貌学造成的扭曲的审慎控制。我们把里曼加速梯度方法描述为我们框架的具体实例。