Beyond spectral gap (extended): The role of the topology in decentralized learning

TL;DR

This paper investigates how the topology of communication graphs affects decentralized learning, revealing limitations of spectral gap theory and providing new insights that align with empirical deep learning results.

Contribution

It introduces a new theoretical framework that captures the impact of graph topology on convergence, surpassing traditional spectral gap analysis.

Findings

Graph topology significantly influences convergence rates.

Spectral gap is not predictive of performance in deep learning.

Collaboration enables larger learning rates than training alone.

Abstract

In data-parallel optimization of machine learning models, workers collaborate to improve their estimates of the model: more accurate gradients allow them to use larger learning rates and optimize faster. In the decentralized setting, in which workers communicate over a sparse graph, current theory fails to capture important aspects of real-world behavior. First, the `spectral gap' of the communication graph is not predictive of its empirical performance in (deep) learning. Second, current theory does not explain that collaboration enables larger learning rates than training alone. In fact, it prescribes smaller learning rates, which further decrease as graphs become larger, failing to explain convergence dynamics in infinite graphs. This paper aims to paint an accurate picture of sparsely-connected distributed optimization. We quantify how the graph topology influences convergence in a quadratic toy problem and provide theoretical results for general smooth and (strongly) convex objectives. Our theory matches empirical observations in deep learning, and accurately describes the relative merits of different graph topologies. This paper is an extension of the conference paper by Vogels et. al. (2022). Code: https://github.com/epfml/topology-in-decentralized-learning.