AI 中文总结
本文提出两种分布式快速不动点算法（ND-DFFP和NI-DFFP），通过Nesterov加速与原始-对偶技术结合，在两种假设下实现复合单调包含问题的$\mathcal{O}(1/k)$收敛速率，并在实验中验证了其高效性。
AI 中文摘要
本文旨在为求解一类单调包含问题开发新的高效分布式算法，该类问题形式为 $0 \in \sum_{i=1}^n (G_ix + T_ix)$，定义在一个由 $n$ 个智能体组成的连通网络上，其中单值算子 $G_i$ 和可能多值的算子 $T_i$ 对智能体 $i$ 保持私有。针对此类问题的现有分布式算法主要未加速，且其在原始原始空间中的精确收敛速率在很大程度上尚未被探索。为弥补这一空白，我们提出了两种基于快速不动点的去中心化算法，即 \texttt{ND-DFFP} 和 \texttt{NI-DFFP}，它们将 Nesterov 型加速与原始-对偶技术相结合，适用于两种显著场景：(i) $G_i$ 的 Lipschitz 连续性以及 $G_i+T_i$ 的极大单调性；(ii) $G_i$ 的余强单调性以及 $T_i$ 的极大单调性。虽然 \texttt{ND-DFFP} 使用同质的依赖于网络的步长，但 \texttt{NI-DFFP} 将问题重新表述为三算子包含，以解耦网络拓扑，从而实现异质的与网络无关的步长。在适当假设下，我们建立了共识误差的 $\mathcal{O}(1/k)$ 收敛速率，以及受限间隙函数和平方前向-后向分裂残差的 $\mathcal{O}(1/k)$ 速率，后两个指标在网络平均迭代或其到有效域的投影上评估。最后，在分布式双线性矩阵博弈和虚拟电厂问题上的数值实验证明了我们的方法相对于文献中近期去中心化基线的竞争性能和计算效率。
英文摘要
This paper aims to develop new and efficient distributed algorithms for solving a class of monotone inclusions, $0 \in \sum_{i=1}^n (G_ix + T_ix)$, over a connected network of $n$ agents, where the single-valued operator $G_i$ and the possibly multivalued operator $T_i$ remain private to agent $i$. Existing distributed algorithms for this problem class are primarily non-accelerated, and their exact convergence rates in the original primal space are largely unexplored. To bridge this gap, we propose two Decentralized Fast Fixed-Point-based algorithms, \texttt{ND-DFFP} and \texttt{NI-DFFP}, which integrate Nesterov-type acceleration with primal-dual techniques under two prominent settings: (i) \textit{Lipschitz continuity of $G_i$ and maximal monotonicity of $G_i+T_i$}; and (ii) \textit{co-coercivity of $G_i$ and maximal monotonicity of $T_i$}. While \texttt{ND-DFFP} utilizes a homogeneous network-dependent stepsize, \texttt{NI-DFFP} reformulates the problem into a three-operator inclusion to decouple the network topology, enabling heterogeneous network-independent stepsizes. Under appropriate assumptions, we establish an $\mathcal{O}(1/k)$ convergence rate for the consensus error and an $\mathcal{O}(1/k)$ rate for both the restricted gap function and the squared forward-backward splitting residual, with the latter two metrics evaluated at the network-average iterate or its projection onto the effective domain. Finally, numerical experiments on distributed bilinear matrix games and a virtual power plant problem demonstrate the competitive performance and computational efficiency of our methods over recent decentralized baselines in the literature.
Comments71 pages, 6 tables, and 2 figures
