Optimal and Practical Algorithms for Smooth and Strongly Convex Decentralized Optimization

TL;DR

This paper introduces two new algorithms for decentralized optimization of smooth strongly convex functions, achieving optimal communication and computation complexities without expensive dual gradient evaluations.

Contribution

The paper presents two novel algorithms that are optimal in communication and gradient computation complexity for decentralized convex optimization, improving upon existing methods.

Findings

First algorithm is optimal in communication rounds and gradient computations.

Second algorithm is optimal in communication rounds without a logarithmic factor.

Numerical experiments demonstrate superior performance over state-of-the-art methods.

Abstract

We consider the task of decentralized minimization of the sum of smooth strongly convex functions stored across the nodes of a network. For this problem, lower bounds on the number of gradient computations and the number of communication rounds required to achieve $\varepsilon$ accuracy have recently been proven. We propose two new algorithms for this decentralized optimization problem and equip them with complexity guarantees. We show that our first method is optimal both in terms of the number of communication rounds and in terms of the number of gradient computations. Unlike existing optimal algorithms, our algorithm does not rely on the expensive evaluation of dual gradients. Our second algorithm is optimal in terms of the number of communication rounds, without a logarithmic factor. Our approach relies on viewing the two proposed algorithms as accelerated variants of the Forward Backward algorithm to solve monotone inclusions associated with the decentralized optimization problem. We also verify the efficacy of our methods against state-of-the-art algorithms through numerical experiments.