Snap-Shot Decentralized Stochastic Gradient Tracking Methods

TL;DR

This paper introduces two new decentralized stochastic gradient tracking algorithms utilizing snapshot techniques, which are more robust to network topology variations and improve convergence properties over existing methods.

Contribution

The paper proposes two novel decentralized stochastic gradient tracking algorithms that are less affected by network topology, enhancing robustness and convergence guarantees compared to prior DSGT methods.

Findings

The new algorithms have improved iteration complexity bounds.

They are more robust to network topology variations.

The convergence is less dependent on network parameters.

Abstract

In decentralized optimization, $m$ agents form a network and only communicate with their neighbors, which gives advantages in data ownership, privacy, and scalability. At the same time, decentralized stochastic gradient descent (\texttt{SGD}) methods, as popular decentralized algorithms for training large-scale machine learning models, have shown their superiority over centralized counterparts. Distributed stochastic gradient tracking~(\texttt{DSGT})~\citep{pu2021distributed} has been recognized as the popular and state-of-the-art decentralized \texttt{SGD} method due to its proper theoretical guarantees. However, the theoretical analysis of \dsgt~\citep{koloskova2021improved} shows that its iteration complexity is $\tilde{\mathcal{O}} \left(\frac{\bar{\sigma}^2}{m\mu \varepsilon} + \frac{\sqrt{L}\bar{\sigma}}{\mu(1 - \lambda_2(W))^{1/2} C_W \sqrt{\varepsilon} }\right)$, where $W$ is a double stochastic mixing matrix that presents the network topology and $ C_W $ is a parameter that depends on $W$. Thus, it indicates that the convergence property of \texttt{DSGT} is heavily affected by the topology of the communication network. To overcome the weakness of \texttt{DSGT}, we resort to the snap-shot gradient tracking skill and propose two novel algorithms. We further justify that the proposed two algorithms are more robust to the topology of communication networks under similar algorithmic structures and the same communication strategy to \dsgt~. Compared with \dsgt, their iteration complexity are $\mathcal{O}\left( \frac{\bar{\sigma}^2}{m\mu\varepsilon} + \frac{\sqrt{L}\bar{\sigma}}{\mu (1 - \lambda_2(W))\sqrt{\varepsilon}} \right)$ and $\mathcal{O}\left( \frac{\bar{\sigma}^2}{m\mu \varepsilon} + \frac{\sqrt{L}\bar{\sigma}}{\mu (1 - \lambda_2(W))^{1/2}\sqrt{\varepsilon}} \right)$ which reduce the impact on network topology (no $C_W$).