2023/06/23 by Yuan, Bo, Fan, Jiaojiao, Liang, Jiaming +2 · 2 citations
#FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2306.13801
We consider the sampling problem from a composite distribution whose potential (negative log density) is ∑i=1n fi(xi)+∑j=1m gj(yj)+∑i=1n∑j=1m\fracσij2η \Vert xi-yj \Vert22 where each of xi and yj is in ℝd, f1, f2, …, fn, g1, g2, …, gm are strongly convex functions, and \σij\ 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 \citelee2021structured. 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 ∝ exp(-∑i=1n fi(x)-∑j=1m gj(x)) in a distributed manner.