We show that, in many settings, the worst-case performance of a distributed optimization algorithm is independent of the number of agents in the system, and can thus be computed in the fundamental case with just two agents. This result relies on a novel approach that systematically exploits symmetries in worst-case performance computation, framed as Semidefinite Programming (SDP) via the Performance Estimation Problem (PEP) framework. Harnessing agent symmetries in the PEP yields compact problems whose size is independent of the number of agents in the system. When all agents are equivalent in the problem, we establish the explicit conditions under which the resulting worst-case performance is independent of the number of agents and is therefore equivalent to the basic case with two agents. Our compact PEP formulation also allows the consideration of multiple equivalence classes of agents, and its size only depends on the number of equivalence classes. This enables practical and automated performance analysis of distributed algorithms in numerous complex and realistic settings, such as the analysis of the worst agent performance. We leverage this new tool to analyze the performance of the EXTRA algorithm in advanced settings and its scalability with the number of agents, providing a tighter analysis and deeper understanding of the algorithm performance.

翻译：我们证明，在许多场景下，分布式优化算法在最坏情况下的性能与系统中的智能体数量无关，因此可以在仅有两个智能体的基本情况下进行计算。这一结果基于一种新颖方法，该方法通过性能估计问题框架将最坏情况性能计算表述为半定规划，并系统性地利用其中的对称性。利用智能体对称性可生成规模与系统中智能体数量无关的紧凑问题。当所有智能体在问题中具有等价性时，我们建立了最坏情况性能与智能体数量无关的显式条件，从而使其等价于两个智能体的基本情形。我们提出的紧凑性能估计问题公式还允许考虑多个智能体等价类，其规模仅取决于等价类的数量。这使得我们能够在大量复杂且现实的场景中（例如最差智能体性能分析）对分布式算法进行实用且自动化的性能分析。我们利用这一新工具分析了EXTRA算法在高级场景中的性能及其随智能体数量的可扩展性，为算法性能提供了更紧致的分析框架和更深入的理解。