We present a methodology to automatically compute worst-case performance bounds for a large class of first-order decentralized optimization algorithms. These algorithms aim at minimizing the average of local functions that are distributed across a network of agents. They typically combine local computations and consensus steps. Our methodology is based on the approach of Performance Estimation Problem (PEP), which allows computing the worst-case performance and a worst-case instance of first-order optimization algorithms by solving an SDP. We propose two ways of representing consensus steps in PEPs, which allow writing and solving PEPs for decentralized optimization. The first formulation is exact but specific to a given averaging matrix. The second formulation is a relaxation but provides guarantees valid over an entire class of averaging matrices, characterized by their spectral range. This formulation often allows recovering a posteriori the worst possible averaging matrix for the given algorithm. We apply our methodology to three different decentralized methods. For each of them, we obtain numerically tight worst-case performance bounds that significantly improve on the existing ones, as well as insights about the parameters tuning and the worst communication networks.

翻译：我们提出了一种方法，能够自动计算一大类一阶去中心化优化算法的最坏情况性能界限。这些算法旨在最小化分布在智能体网络中的局部函数的平均值，通常结合了局部计算与一致性步骤。我们的方法基于性能估计问题（PEP）框架，该框架通过求解半定规划来为一阶优化算法计算最坏情况性能及其对应的最坏实例。我们提出了两种在PEP中表示一致性步骤的方式，从而能够编写并求解去中心化优化的PEP。第一种表述是精确的，但仅适用于给定的平均矩阵。第二种表述是一种松弛形式，但能保证其有效性覆盖由谱范围刻画的一整类平均矩阵。该表述通常能事后恢复出给定算法的最差平均矩阵。我们将该方法应用于三种不同的去中心化方法。对于每种方法，我们获得了数值上紧致的最坏情况性能界限，这些界限显著优于现有结果，同时提供了关于参数调优和最差通信网络的见解。