The Nesterov accelerated gradient (NAG) method is an important extrapolation-based numerical algorithm that accelerates the convergence of the gradient descent method in convex optimization. When dealing with an objective function that is $\mu$-strongly convex, selecting extrapolation coefficients dependent on $\mu$ enables global R-linear convergence. In cases where $\mu$ is unknown, a commonly adopted approach is to set the extrapolation coefficient using the original NAG method. This choice allows for achieving the optimal iteration complexity among first-order methods for general convex problems. However, it remains unknown whether the NAG method with an unknown strongly convex parameter exhibits global R-linear convergence for strongly convex problems. In this work, we answer this question positively by establishing the Q-linear convergence of certain constructed Lyapunov sequences. Furthermore, we extend our result to the global R-linear convergence of the accelerated proximal gradient method, which is employed for solving strongly convex composite optimization problems. Interestingly, these results contradict the findings of the continuous counterpart of the NAG method in [Su, Boyd, and Cand\'es, J. Mach. Learn. Res., 2016, 17(153), 1-43], where the convergence rate by the suggested ordinary differential equation cannot exceed the $O(1/{\tt poly}(k))$ for strongly convex functions.

翻译：Nesterov加速梯度(NAG)方法是一种重要的基于外推的数值算法，可加速凸优化中梯度下降法的收敛速度。当目标函数为$\mu$-强凸时，选取依赖于$\mu$的外推系数可实现全局R-线性收敛。在$\mu$未知的情况下，常用做法是采用原始NAG方法设定外推系数，这一选择使得一般凸问题中一阶方法的迭代复杂度达到最优。然而，对于强凸问题，未知强凸参数条件下的NAG方法是否具有全局R-线性收敛性仍属未知。本文通过建立特定Lyapunov序列的Q-线性收敛性，对此问题给出了肯定回答。进一步地，我们将结果推广至用于求解强凸复合优化问题的加速近端梯度法的全局R-线性收敛性。有趣的是，这些结论与[Su, Boyd, and Candès, J. Mach. Learn. Res., 2016, 17(153), 1-43]中NAG方法连续时间对应体的研究结果相矛盾——该文献通过建议的常微分方程所获得的收敛速率在强凸函数情形下无法超越$O(1/{\tt poly}(k))$。