论文标题
地球上凸优化的加速梯度方法：可处理算法和收敛分析
Accelerated Gradient Methods for Geodesically Convex Optimization: Tractable Algorithms and Convergence Analysis
论文作者
论文摘要
我们提出了可用于riemannian优化的计算处理的一阶方法，从而扩展了Nesterov加速梯度（NAG）方法。对于地球凸和地理上强烈凸的物镜的均具有相同的算法，仅在仅在标准假设下，我们的算法与欧几里得空间上的NAG方法具有相同的迭代复杂性。据我们所知，提出的方案是第一种完全加速的方法，用于凸出凸优化问题。我们的收敛分析利用了新型度量失真引理以及精心设计的潜在功能。还通过让步骤尺寸倾向于零来识别与（Alimisis等，2020）中的Riemannian加速度建模的连续时间动力学的连接。我们通过数值实验来验证理论结果。
We propose computationally tractable accelerated first-order methods for Riemannian optimization, extending the Nesterov accelerated gradient (NAG) method. For both geodesically convex and geodesically strongly convex objective functions, our algorithms are shown to have the same iteration complexities as those for the NAG method on Euclidean spaces, under only standard assumptions. To the best of our knowledge, the proposed scheme is the first fully accelerated method for geodesically convex optimization problems. Our convergence analysis makes use of novel metric distortion lemmas as well as carefully designed potential functions. A connection with the continuous-time dynamics for modeling Riemannian acceleration in (Alimisis et al., 2020) is also identified by letting the stepsize tend to zero. We validate our theoretical results through numerical experiments.
