The optimized gradient method (OGM) provides a factor-\(\sqrt{2}\) speedup upon Nesterov’s celebrated accelerated gradient method in the convex (but non-strongly convex) setup. However, this improved acceleration mechanism has not been well understood; prior analyses of OGM relied on a computer-assisted proof methodology, so the proofs were opaque for humans despite being verifiable and correct. In this work, we present a new analysis of OGM based on a Lyapunov function and linear coupling. These analyses are developed and presented without the assistance of computers and are understandable by humans. Furthermore, we generalize OGM’s acceleration mechanism and obtain a factor-\(\sqrt{2}\) speedup in other setups: acceleration with a simpler rational stepsize, the strongly convex setup, and the mirror descent setup.

优化梯度法 (OGM) 比Nesterov 在凸（但非强凸）设置中著名的加速梯度法提供了\(\sqrt{2}\)倍的加速。然而，这种改进的加速机制尚未得到很好的理解；之前对 OGM 的分析依赖于计算机辅助证明方法，因此尽管证明是可验证且正确的，但对人类来说是不透明的。在这项工作中，我们提出了一种基于 Lyapunov 函数和线性耦合的 OGM 的新分析。这些分析是在没有计算机帮助的情况下开发和呈现的，并且是人类可以理解的。此外，我们推广了 OGM 的加速机制并获得了一个因子 - \(\sqrt{2}\)其他设置中的加速：使用更简单的合理步长加速、强凸设置和镜像下降设置。