DoCoM: Compressed Decentralized Optimization with Near-Optimal Sample Complexity

TL;DR

This paper introduces DoCoM, a communication-efficient decentralized optimization algorithm that combines momentum and gradient tracking with compression, achieving near-optimal sample complexity and fast convergence.

Contribution

The paper presents DoCoM, a novel algorithm that integrates momentum and compressed consensus to improve efficiency in decentralized optimization.

Findings

Achieves near-stationary solutions with $ ext{O}(1/T^{2/3})$ convergence rate.

Proves linear convergence under Polyak-Łojasiewicz condition.

Outperforms existing algorithms in numerical experiments.

Abstract

This paper proposes the Doubly Compressed Momentum-assisted stochastic gradient tracking algorithm $\texttt{DoCoM}$ for communication-efficient decentralized optimization. The algorithm features two main ingredients to achieve a near-optimal sample complexity while allowing for communication compression. First, the algorithm tracks both the averaged iterate and stochastic gradient using compressed gossiping consensus. Second, a momentum step is incorporated for adaptive variance reduction with the local gradient estimates. We show that $\texttt{DoCoM}$ finds a near-stationary solution at all participating agents satisfying $\mathbb{E}[ \| \nabla f( \theta ) \|^2 ] = \mathcal{O}( 1 / T^{2/3} )$ in $T$ iterations, where $f(\theta)$ is a smooth (possibly non-convex) objective function. Notice that the proof is achieved via analytically designing a new potential function that tightly tracks the one-iteration progress of $\texttt{DoCoM}$. As a corollary, our analysis also established the linear convergence of $\texttt{DoCoM}$ to a global optimal solution for objective functions with the Polyak-{\L}ojasiewicz condition. Numerical experiments demonstrate that our algorithm outperforms several state-of-the-art algorithms in practice.