We consider the sampling problem from a composite distribution whose potential (negative log density) is $\sum_{i=1}^n f_i(x_i)+\sum_{j=1}^m g_j(y_j)+\sum_{i=1}^n\sum_{j=1}^m\frac{\sigma_{ij}}{2\eta} \Vert x_i-y_j \Vert^2_2$ where each of $x_i$ and $y_j$ is in $\mathbb{R}^d$, $f_1, f_2, \ldots, f_n, g_1, g_2, \ldots, g_m$ are strongly convex functions, and $\{\sigma_{ij}\}$ encodes a network structure. % motivated by the task of drawing samples over a network in a distributed manner. Building on the Gibbs sampling method, we develop an efficient sampling framework for this problem when the network is a bipartite graph. More importantly, we establish a non-asymptotic linear convergence rate for it. This work extends earlier works that involve only a graph with two nodes \cite{lee2021structured}. To the best of our knowledge, our result represents the first non-asymptotic analysis of a Gibbs sampler for structured log-concave distributions over networks. Our framework can be potentially used to sample from the distribution $ \propto \exp(-\sum_{i=1}^n f_i(x)-\sum_{j=1}^m g_j(x))$ in a distributed manner.

翻译：我们考虑从一个复合分布中采样的问题，该分布的势函数（负对数密度）为 $\sum_{i=1}^n f_i(x_i)+\sum_{j=1}^m g_j(y_j)+\sum_{i=1}^n\sum_{j=1}^m\frac{\sigma_{ij}}{2\eta} \Vert x_i-y_j \Vert^2_2$，其中每个 $x_i$ 和 $y_j$ 属于 $\mathbb{R}^d$，$f_1, f_2, \ldots, f_n, g_1, g_2, \ldots, g_m$ 是强凸函数，且 $\{\sigma_{ij}\}$ 编码了一个网络结构。基于吉布斯采样方法，当该网络为二分图时，我们为该问题开发了一个高效的采样框架。更重要的是，我们建立了其非渐近线性收敛速率。本工作扩展了早期仅涉及两个节点图的研究 \cite{lee2021structured}。据我们所知，我们的结果首次实现了对网络上结构化对数凹分布的吉布斯采样的非渐近分析。该框架可潜在用于以分布式方式从分布 $ \propto \exp(-\sum_{i=1}^n f_i(x)-\sum_{j=1}^m g_j(x))$ 中采样。