We study the communication complexity of convex decentralized optimization over time-varying networks, where $n$ nodes hold private functions and must agree on the global minimizer using only synchronous exchanges with neighbors. The cost is the number of communication rounds to reach accuracy $\varepsilon$ -- a measure akin to round complexity in the LOCAL model, but constrained by nodes sharing only oracle responses. We prove a new lower bound of $Ω\!\left(χ_{\mathcal G} \sqrt{κ_g}\,\log\frac{n}{χ_{\mathcal G}}\log\frac1\varepsilon\right)$ communication rounds, where $χ_{\mathcal G}$ is the condition number of the network Laplacians and $κ_g$ that of the global objective, showing the round complexity attainable under uniform regularity cannot be matched in the nonuniform regime. The construction rests on spectral graph theory: we embed time-rotating star gadgets into the edges of an expander and patch them to preserve spectral connectivity.

翻译：我们研究时变网络上凸去中心化优化的通信复杂度，其中n个节点持有私有函数，必须仅通过与邻居的同步交换来达成全局极小值的一致。成本是达到精度ε所需的通信轮数——这一度量类似于LOCAL模型中的轮复杂度，但受限于节点仅共享预言机响应。我们证明了一个新的下界Ω(χ𝒢 √κ_g log(n/χ𝒢) log(1/ε))通信轮数，其中χ𝒢是网络拉普拉斯矩阵的条件数，κ_g是全局目标函数的条件数，这表明在均匀正则性下可达到的轮复杂度在非均匀情形下无法匹配。该构造基于谱图理论：我们将时间旋转星形结构嵌入到扩展图的边中，并通过修补来保持谱连通性。