We propose a new unifying framework, Birch SGD, for analyzing and designing distributed SGD methods. The central idea is to represent each method as a weighted directed tree, referred to as a computation tree. Leveraging this representation, we introduce a general theoretical result that reduces convergence analysis to studying the geometry of these trees. This perspective yields a purely graph-based interpretation of optimization dynamics, offering a new and intuitive foundation for method development. Using Birch SGD, we design eight new methods and analyze them alongside previously known ones, with at least six of the new methods shown to have optimal computational time complexity. Our research leads to two key insights: (i) all methods share the same "iteration rate" of $O\left(\frac{(R + 1) L Δ}{\varepsilon} + \frac{σ^2 L Δ}{\varepsilon^2}\right)$, where $R$ the maximum "tree distance" along the main branch of a tree; and (ii) different methods exhibit different trade-offs-for example, some update iterates more frequently, improving practical performance, while others are more communication-efficient or focus on other aspects. Birch SGD serves as a unifying framework for navigating these trade-offs. We believe these results provide a unified foundation for understanding, analyzing, and designing efficient asynchronous and parallel optimization methods.

翻译：我们提出了一种新的统一框架——Birch SGD，用于分析和设计分布式随机梯度下降（SGD）方法。其核心思想是将每种方法表示为一棵加权有向树，称为计算树。借助这一表示，我们引入了一个通用理论结果，将收敛性分析简化为研究这些树的几何结构。这一视角为优化动力学提供了纯粹的图论解释，为方法开发奠定了新颖且直观的基础。利用Birch SGD，我们设计了八种新方法，并与已知方法一同进行了分析，其中至少六种新方法被证明具有最优计算时间复杂度。我们的研究得出两个关键见解：（i）所有方法共享相同的“迭代速率”$O\left(\frac{(R + 1) L Δ}{\varepsilon} + \frac{σ^2 L Δ}{\varepsilon^2}\right)$，其中$R$为树主分支上的最大“树距离”；（ii）不同方法展现出不同的权衡——例如，某些方法更新迭代更频繁，从而改善实际性能，而另一些方法则更具通信效率或侧重于其他方面。Birch SGD作为导航这些权衡的统一框架。我们相信，这些结果为理解、分析和设计高效异步及并行优化方法提供了统一基础。