发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于消息传递的图分解框架，通过联合设计优化子问题和通信实现去中心化优化，并给出首个实例GATE及其轻量变体GATE-S，证明了线性收敛性并验证了效率。
AI 中文摘要
我们研究在无向图上对光滑强凸函数之和的最小化问题，其中每个函数由一个智能体持有，通信仅限于图中的邻居。现有的去中心化方法，无论是基于八卦（gossip）还是基于生成树上的路由，通常利用网络来混合或聚合信息，以实现{\it 预设的}局部优化更新。这种以通信为中心的观点所缺乏的是一个通用框架，该框架利用图结构来{\it 联合}设计优化子问题以及智能体通过协作计算和通信来共同解决这些子问题。我们从第一性原理出发开发了这样一个框架，联合设计了一致性约束的线性表示、所得对偶变量的块（联合优化），以及协同解决每个块子问题的连通智能体簇，这些智能体在指定的子图上协作求解。GATE（图撕裂消息传递）是该框架的第一个实例：每条边一个变量，树块。在每次迭代中，智能体通过最小化两个端点代价到目标消息之和并松弛结果来更新其分配的边变量。消息通过遵循树递归的局部最小化来更新。为了降低每次迭代的计算和通信成本，我们开发了GATE-S，一种使用易处理的局部模型和轻量级消息参数化的代理变体。我们建立了线性收敛性，其速率明确地依赖于函数正则性、网络拓扑和所选划分之间的相互作用，揭示了图分解的影响。进行了数值实验以验证理论结果并评估我们算法的效率。
英文摘要
We study the minimization of sums of smooth strongly convex functions over undirected graphs, with each function held by one agent and communication restricted to neighbors in the graph. Existing decentralized methods, whether based on gossip or on routing over spanning trees, typically use the network to mix or aggregate information to enable {\it prescribed} local optimization updates. What this communication-centered viewpoint lacks is a general framework that uses graph structure to {\it jointly} design the optimization subproblems and the cooperative computation and communication through which agents solve them cooperatively. We develop such a framework from first principles, jointly designing the linear representation of agreement constraints, the blocks of the resulting dual variables (jointly optimized), and connected cluster of agents that cooperatively solve each block subproblem over the assigned subgraph. GATE (Graph-Tearing message passing) is a first instance of this framework: one variable per edge and tree blocks. At each iteration, agents update their assigned edge variables by minimizing the sum of the two endpoint cost-to-go messages and relaxing the result. The messages are updated through local minimizations following the tree recursion. To reduce per-iteration computational and communication costs, we develop GATE-S, a surrogate variant using tractable local models and lightweight message parametrizations. We establish linear convergence with a rate explicit in the interplay among function regularity, network topology, and the chosen partition, revealing the effects of graph decomposition. Numerical experiments are conducted to validate the theoretical results and evaluate the efficiency of our algorithms.
