Birch SGD: A Tree Graph Framework for Local and Asynchronous SGD Methods

TL;DR

Birch SGD introduces a graph-based framework representing distributed SGD methods as computation trees, enabling unified analysis, new method design, and insights into trade-offs like update frequency and communication efficiency.

Contribution

The paper presents Birch SGD, a novel tree graph framework for analyzing and designing distributed SGD methods, leading to new algorithms and a unified understanding of their dynamics.

Findings

Eight new methods designed using Birch SGD.

At least six methods achieve optimal computational complexity.

All methods share the same iteration rate, with trade-offs in update frequency and communication.

Abstract

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 $O\left(\frac{(R + 1) L \Delta}{\varepsilon} + \frac{\sigma^2 L \Delta}{\varepsilon^2}\right)$, 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.