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，它在凸优化中加速了梯度下降法的收敛。当处理目标函数为$\mu$-强凸时，选择依赖于$\mu$的外推系数可实现全局R线性收敛。在$\mu$未知的情况下，通常采用的方法是使用原始NAG方法设置外推系数。这一选择使得算法能在一般凸问题的一阶方法中达到最优迭代复杂度。然而，对于强凸问题，强凸参数未知的NAG方法是否具有全局R线性收敛性仍属未知。在本工作中，我们通过建立特定构造的李雅普诺夫序列的Q线性收敛性，对此问题给出了肯定的回答。此外，我们将结果推广到加速近端梯度方法的全局R线性收敛性，该方法用于求解强凸复合优化问题。有趣的是，这些结果与NAG方法的连续对应模型[Su, Boyd, and Cand\'es, J. Mach. Learn. Res., 2016, 17(153), 1-43]中的发现相矛盾，在该模型中，针对强凸函数，所建议的常微分方程给出的收敛速率无法超过$O(1/{\tt poly}(k))$。