Tight bounds on the convergence rate of generalized ratio consensus algorithms

The problems discussed in this paper are motivated by general ratio consensus algorithms, introduced by Kempe, Dobra, and Gehrke (2003) in a simple form as the push-sum algorithm, later extended by B\'en\'ezit et al. (2010) under the name weighted gossip algorithm. We consider a communication protocol described by a strictly stationary, ergodic, sequentially primitive sequence of non-negative matrices, applied iteratively to a pair of fixed initial vectors, the components of which are called values and weights defined at the nodes of a network. The subject of ratio consensus problems is to study the asymptotic properties of ratios of values and weights at each node, expecting convergence to the same limit for all nodes. The main results of the paper provide upper bounds for the rate of the almost sure exponential convergence in terms of the spectral gap associated with the given sequence of random matrices. It will be shown that these upper bounds are sharp. Our results complement previous results of Picci and Taylor (2013) and Iutzeler, Ciblat and Hachem (2013)

Mixing time of an unaligned Gibbs sampler on the square

The paper concerns a particular example of the Gibbs sampler and its mixing efficiency. Coordinates of a point are rerandomized in the unit square $[0,1]^2$ to approach a stationary distribution with density proportional to $\exp(-A^2(u-v)^2)$ for $(u,v)\in [0,1]^2$ with some large parameter $A$. Diaconis conjectured the mixing time of this process to be $O(A^2)$ which we confirm in this paper. This improves on the currently known $O(\exp(A^2))$ estimate

Analysis of a non-reversible Markov chain speedup by a single edge

We present a Markov chain example where non-reversibility and an added edge jointly improve mixing time: when a random edge is added to a cycle of $n$ vertices and a Markov chain with a drift is introduced, we get mixing time of $O(n^{3/2})$ with probability bounded away from 0. If only one of the two modifications were performed, the mixing time would stay $\Omega(n^2)$.Comment: 15 pages, 5 figure

Improved mixing rates of directed cycles by added connection

We investigate the mixing rate of a Markov chain where a combination of long distance edges and non-reversibility is introduced: as a first step, we focus here on the following graphs: starting from the cycle graph, we select random nodes and add all edges connecting them. We prove a square factor improvement of the mixing rate compared to the reversible version of the Markov chain

Push sum with transmission failures

The push-sum algorithm allows distributed computing of the average on a directed graph, and is particularly relevant when one is restricted to one-way and/or asynchronous communications. We investigate its behavior in the presence of unreliable communication channels where messages can be lost. We show that exponential convergence still holds and deduce fundamental properties that implicitly describe the distribution of the final value obtained. We analyze the error of the final common value we get for the essential case of two nodes, both theoretically and numerically. We provide performance comparison with a standard consensus algorithm

Computable convergence rate bound for ratio consensus algorithms

The objective of the paper is to establish a computable upper bound on the almost sure convergence rate for a class of ratio consensus algorithms. Our result extends the works of Iutzeler et al. (2013) on similar bounds that have been obtained in a more restrictive setup with limited conclusions. It also complements the results of Gerencs\'er and Gerencs\'er (2021) that identified the exact convergence rate which is however not computable in general

On the Poisson Equation of Parameter-Dependent Markov Chains

The objective of the paper is to revisit a key mathematical technology within the theory of stochastic approximation in a Markovian framework, elaborated in much detail by Benveniste, M\'etivier, and Priouret (1990): the existence, uniqueness and Lipschitz-continuity of the solutions of parameter-dependent Poisson equations associated with parameter-dependent Markov chains on general state spaces. The setup and the methodology of our investigation is based on a new, elegant stability theory for Markov chains, developed by Hairer and Mattingly (2011)

Correlation bound for distant parts of factor of IID processes

We study factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most $k(d-1) / (\sqrt{d-1})^k$, where $k$ denotes the distance of the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.Comment: 18 pages, 5 figure

Low complexity convergence rate bounds for the synchronous gossip subclass of push-sum algorithms

We develop easily accessible quantities for bounding the almost sure exponential convergence rate of push-sum algorithms. We analyze the scenario of i.i.d. synchronous gossip, every agent communicating towards its single target at every step. Multiple bounding expressions are developed depending on the generality of the setup, all functions of the spectrum of the network. While the most general bound awaits further improvement, with more symmetries, close bounds can be established, as demonstrated by numerical simulations.Comment: 15 pages, 8 figure