We establish that in distributed optimization, the prevalent strategy of minimizing the second-largest eigenvalue modulus (SLEM) of the averaging matrix for selecting communication weights, while optimal for existing theoretical performance bounds, is generally not optimal regarding the exact worst-case performance of the algorithms. This exact performance can be computed using the Performance Estimation Problem (PEP) approach. We thus rely on PEP to formulate an optimization problem that determines the optimal communication weights for a distributed optimization algorithm deployed on a specified undirected graph. Our results show that the optimal weights can outperform the weights minimizing the second-largest eigenvalue modulus (SLEM) of the averaging matrix. This suggests that the SLEM is not the best characterization of weighted network performance for decentralized optimization. Additionally, we explore and compare alternative heuristics for weight selection in distributed optimization.

翻译：我们证明，在分布式优化中，尽管选择使平均矩阵的第二大特征值模（SLEM）最小化的通信权重这一主流策略对于现有理论性能界而言是最优的，但该策略通常并非算法精确最坏情况性能的最优选择。通过使用性能估计问题（PEP）方法可计算这种精确性能。据此，我们利用PEP构造了一个优化问题，用于确定在指定无向图上部署的分布式优化算法的最优通信权重。结果表明，最优权重的性能可超越使平均矩阵的第二大特征值模（SLEM）最小化的权重，这意味着SLEM并非分布式最优化的加权网络性能的最佳表征指标。此外，我们还探索并比较了分布式优化中替代性的权重选取启发式方法。