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，分布式优化算法的最坏性能与系统中智能体的数量无关，因此可以在仅有两个智能体的基础情形中进行计算。这一结果依赖于一种新颖方法，该方法系统性地利用了最坏性能计算中的对称性，并通过性能估计问题（PEP）框架将其表述为半定规划（SDP）。在PEP中利用智能体对称性可得到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等价时，我们明确了所得最坏性能与智能体数量无关的条件，从而等价于仅有两个智能体的基础情形。我们的紧凑PEP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。这使得在众多复杂且现实的场景中（例如分析最差智能体性能）对分布式算法进行实用且自动化的性能分析成为可能。我们利用这一新工具分析了EXTRA算法在高级设置中的性能及其随智能体数量的可扩展性，提供了更严格的分析和对算法性能的深入理解。