First-order methods play a central role in large-scale convex optimization. Despite their various forms of descriptions and many applications, such methods mostly and fundamentally rely on two basic types of analyses: gradient-descent analysis, which yields primal progress, and mirror-descent analysis, which yields dual progress. In this paper, we observe that the performances of these two analyses are complementary, so that faster algorithms can be designed by coupling the two analyses, and their corresponding descent steps. In particular, we show in this paper how to obtain a conceptually simple reinterpretation of Nesterov's accelerated gradient method [Nes83, Nes04, Nes05]. Nesterov's method is the optimal first-order method for the class of smooth convex optimization problems. However, the proof of the fast convergence of Nesterov's method has no clear interpretation and is regarded by some as relying on an `algebraic trick'. We apply our novel insights to express Nesterov's algorithm as a coupling of gradient descent and mirror descent, and as a result, the convergence proof can be understood as some natural combination of the two underlying convergence analyses. We believe that this complementary view of the two types of analysis may not only facilitate the study of Nesterov's method in a white-box manner so as to apply it to problems outside its original scope, but also let us design better first-order methods in a conceptually easier way.

第一阶段方法在大规模机器学习中扮演了核心作用，在保持渐进进展和提供双重进展的基础上，我们观察到梯度和镜像下降的性能是互补的，因此可以通过将两者进行线性耦合来设计更快速的算法，我们展示了如何使用线性耦合重构Nesterov的加速梯度方法，并通过将其扩展到Nesterov方法无法适用的许多其他设置来讨论线性耦合的威力。

线性耦合：梯度下降与镜像下降的终极统一