An Accelerated Distributed Stochastic Gradient Method with Momentum

TL;DR

This paper presents DSMT, an accelerated distributed stochastic gradient method with momentum that achieves near-centralized convergence rates with minimal communication, suitable for large-scale networked optimization.

Contribution

Introduces DSMT, a single-loop distributed stochastic gradient method with momentum and Chebyshev acceleration, achieving optimal convergence rates with minimal communication.

Findings

Achieves convergence rates comparable to centralized SGD.

Transient times scale as (n^{5/3}/(1-)), optimal among existing methods.

Does not require multiple inter-node communications or gradient accumulation.

Abstract

In this paper, we introduce an accelerated distributed stochastic gradient method with momentum for solving the distributed optimization problem, where a group of $n$ agents collaboratively minimize the average of the local objective functions over a connected network. The method, termed ``Distributed Stochastic Momentum Tracking (DSMT)'', is a single-loop algorithm that utilizes the momentum tracking technique as well as the Loopless Chebyshev Acceleration (LCA) method. We show that DSMT can asymptotically achieve comparable convergence rates as centralized stochastic gradient descent (SGD) method under a general variance condition regarding the stochastic gradients. Moreover, the number of iterations (transient times) required for DSMT to achieve such rates behaves as $\mathcal{O}(n^{5/3}/(1-\lambda))$ for minimizing general smooth objective functions, and $\mathcal{O}(\sqrt{n/(1-\lambda)})$ under the Polyak-{\L}ojasiewicz (PL) condition. Here, the term $1-\lambda$ denotes the spectral gap of the mixing matrix related to the underlying network topology. Notably, the obtained results do not rely on multiple inter-node communications or stochastic gradient accumulation per iteration, and the transient times are the shortest under the setting to the best of our knowledge.