# Analysis and Design of First-Order Distributed Optimization Algorithms over Time-Varying Graphs

## TL;DR

This paper provides a unified analysis framework for first-order distributed optimization algorithms over time-varying graphs, introducing a new algorithm called SVL with improved convergence.

## Contribution

It offers a computationally efficient analysis method and proposes the SVL algorithm that outperforms existing methods in convergence speed.

## Key findings

- Unified analysis yields worst-case linear convergence rate
- Analysis framework involves a small fixed-size semidefinite program
- SVL algorithm achieves faster convergence than existing algorithms

## Abstract

This work concerns the analysis and design of distributed first-order optimization algorithms over time-varying graphs. The goal of such algorithms is to optimize a global function that is the average of local functions using only local computations and communications. Several different algorithms have been proposed that achieve linear convergence to the global optimum when the local functions are strongly convex. We provide a unified analysis that yields the worst-case linear convergence rate as a function of the condition number of the local functions, the spectral gap of the graph, and the parameters of the algorithm. The framework requires solving a small semidefinite program whose size is fixed; it does not depend on the number of local functions or the dimension of their domain. The result is a computationally efficient method for distributed algorithm analysis that enables the rapid comparison, selection, and tuning of algorithms. Finally, we propose a new algorithm, which we call SVL, that is easily implementable and achieves a faster worst-case convergence rate than all other known algorithms.