AI 中文总结
研究给出DIGing和AugDGM两种分布式梯度跟踪算法的精确最坏情况收敛速率，通过特征分解简化分析，得出显式公式并发现图连通性拐点，数值实验验证理论，揭示不同连通性下算法与集中式梯度下降最优收敛速率的关系。
AI 中文摘要
与大多数现有文献在分析分布式优化算法时报告充分收敛条件并导致保守收敛速率不同，本文给出了两种典型梯度跟踪算法DIGing和AugDGM的精确最坏情况收敛速率。通过特征分解表明两种算法平均状态动态相同，但梯度跟踪子系统不同，而该子系统决定算法收敛。利用分解的对角结构将多输入多输出系统的稳定性分析简化为一组参数变化的单输入单输出系统，从而得出精确最坏情况收敛速率的显式公式。公式表明在相同假设下DIGing的最优最坏情况收敛速率大于AugDGM。还发现图连通性存在拐点σ = 1/3，连通性优于此拐点时，AugDGM在目标函数条件数足够差时可实现集中式梯度下降的最优收敛速率，反之则无法实现。数值实验验证了理论结果。
英文摘要
Different from most existing literature in the analysis of distributed optimization algorithms that reports sufficient convergence conditions leading to a conservative convergence rate, this work provides the exact worst-case convergence rates for two typical gradient tracking algorithms, DIGing and AugDGM. By eigen-decomposition, we show that two algorithms share the same average-state dynamics, while they differ from each other in the gradient tracking subsystems, which entirely govern algorithm convergence. Exploiting the diagonal structure of this decomposition, we reduce the stability analysis of MIMO systems to that of a set of parameter-varying SISO systems, from which explicit formulas for the exact worst-case convergence rate can be derived. These formulas clearly show that the optimal worst-case convergence rate of DIGing is larger than that of AugDGM under the same assumptions on objective functions and communication networks. Furthermore, we find that there is an inflection point in the graph connectivity, which is $σ=1/3$. For graphs with connectivity better than this inflection point, the optimal convergence rate of centralized gradient descent can be achieved by AugDGM provided the condition number of objective functions is worse enough. On the other hand, for graphs with connectivity worse than this inflection point, the centralized optimal rate can never be achieved. Numerical experiments validate the theoretical results.
