AI 中文总结
研究分布式优化算法DIGing在最小化非加权平方和时的最优参数设计，通过将迭代算法表示为动态线性系统并分解，利用劳斯稳定性判据推导最优收敛速率及参数公式，为DIGing算法最优参数设计迈出第一步。
AI 中文摘要
在分布式优化算法中，没有通用方法来设计合适参数以实现更快收敛。本文考虑目标函数为非加权平方和的分布式不精确梯度跟踪（DIGing）算法。通过将迭代算法表示为动态线性系统，分解为不同图频率并得到一组解耦子系统，便于分析收敛速率。利用控制理论中的劳斯稳定性判据，推导了最优最坏情况收敛速率及相应参数的显式公式。发现DIGing即使对最简单目标函数收敛速率也慢，该方法是DIGing算法求解一般目标函数最优参数设计的第一步。
英文摘要
There is no general method for designing proper parameters to achieve faster convergence in distributed optimization algorithms. In this paper, we consider the distributed inexact gradient tracking (DIGing) algorithm with the objective function being the unweighted sum of squares. By representing the iteration algorithm as a dynamical linear system, we decompose it into different graph frequencies and obtain a set of decoupled subsystems, on which we can easily analyze the convergence rate. By using Routh stability criterion from control theory, we derive the explicit formula of the optimal worst-case convergence rate and the corresponding parameters. We can see that the convergence rate of DIGing is slow even for the simplest objective functions, thus acceleration is necessary for general application. The proposed method can be viewed as the first step toward optimal parameter design of DIGing algorithm in solving general objective functions.
