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.
Categories: cs.LG cs.DC cs.DS math.OC