Lower Bounds and Optimal Algorithms for Non-Smooth Convex Decentralized Optimization over Time-Varying Networks

TL;DR

This paper establishes the first lower bounds and develops an optimal algorithm for non-smooth convex decentralized optimization over time-varying networks, addressing a key gap in the theoretical understanding of such problems.

Contribution

It provides the first lower bounds and an optimal algorithm for non-smooth decentralized optimization over dynamic networks, advancing theoretical foundations.

Findings

First lower bounds on communication and computation complexities.

An optimal algorithm matching the lower bounds.

Significant theoretical performance improvements over existing methods.

Abstract

We consider the task of minimizing the sum of convex functions stored in a decentralized manner across the nodes of a communication network. This problem is relatively well-studied in the scenario when the objective functions are smooth, or the links of the network are fixed in time, or both. In particular, lower bounds on the number of decentralized communications and (sub)gradient computations required to solve the problem have been established, along with matching optimal algorithms. However, the remaining and most challenging setting of non-smooth decentralized optimization over time-varying networks is largely underexplored, as neither lower bounds nor optimal algorithms are known in the literature. We resolve this fundamental gap with the following contributions: (i) we establish the first lower bounds on the communication and subgradient computation complexities of solving non-smooth convex decentralized optimization problems over time-varying networks; (ii) we develop the first optimal algorithm that matches these lower bounds and offers substantially improved theoretical performance compared to the existing state of the art.