Convergence of block coordinate descent with diminishing radius for nonconvex optimization

Block coordinate descent (BCD), also known as nonlinear Gauss-Seidel, is a simple iterative algorithm for nonconvex optimization that sequentially minimizes the objective function in each block coordinate while the other coordinates are held fixed. We propose a version of BCD that is guaranteed to converge to the stationary points of block-wise convex and differentiable objective functions under constraints. Furthermore, we obtain a best-case rate of convergence of order $\log n/\sqrt{n}$, where $n$ denotes the number of iterations. A key idea is to restrict the parameter search within a diminishing radius to promote stability of iterates, and then to show that such auxiliary constraints vanish in the limit. As an application, we provide a modified alternating least squares algorithm for nonnegative CP tensor factorization that converges to the stationary points of the reconstruction error with the same bound on the best-case rate of convergence. We also experimentally validate our results with both synthetic and real-world data.Comment: 12 pages, 2 figure. Rate of convergence added. arXiv admin note: text overlap with arXiv:2009.0761

Sampling random graph homomorphisms and applications to network data analysis

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph $F$ into a large network $\mathcal{G}$. We propose two complementary MCMC algorithms for sampling a random graph homomorphisms and establish bounds on their mixing times and concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neigborhood sampling. Various time averages of the MCMC trajectory give us various computable observables, including well-known ones such as homomorphism density and average clustering coefficient and their generalizations. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We provide various examples and simulations demonstrating our framework through synthetic networks. We also apply our framework for network clustering and classification problems using the Facebook100 dataset and Word Adjacency Networks of a set of classic novels.Comment: 51 pages, 33 figures, 2 table