We study two families of online convex optimization algorithms: mirror descent and follow-the-regularized-leader. We prove that many mirror descent algorithms (such as online gradient descent) are actually instances of a more general follow-the-leader algorithm which uses proximal regularization (additional strong convexity centered at the current feasible point). Further, under certain step-size assumptions, other mirror-descent algorithms are equivalent to follow-the-leader algorithms with origin-centered regularization (such as dual averaging). Building on these observations, we provide a general analysis for the case of L2 regularization that accommodates general implicit updates as well as composite objectives. This analysis tightens (by a constant factor) and generalizes earlier analysis of the follow-the-proximally-regularized-leader algorithm.

该研究探讨了三家在线凸优化算法家族：follow-the-proximally-regularized-leader（FTRL-Proximal）、正则化双平均（RDA）和组合目标镜像下降。研究证明了所有这些算法都是通用FTRL更新的实例。此外，通过使用紧凑的表示方法，文中还提出了一种更好的算法性能估计方法，在真实数据集上展现出了更好的性能。

带隐式更新的正则化对偶平均与镜像下降的统一视角