Column-stochastic Matrices Research Articles

Distributed online convex optimization with multiple coupled constraints: A double accelerated push–pull algorithm

This paper is concerned with the distributed online convex optimization problems with multiple coupled constraints over unbalanced digraphs, which are widely applied in diverse fields such as machine learning, smart grids, and resource allocation problems, etc. In such practical scenarios, multiple coupled constraints, where each coupled constraint might only contain a portion of all nodes, are more general and complex than global coupled constraints. Additionally, due to certain constraint conditions such as energy and communication limitations, each node can only exchange information with other nodes in an identical constraint. Besides, the network connectivity of practical constrained problems might not be only confined to one type of weight matrices. By adopting an augmented Lagrangian method, the multiple coupled constrained optimization problem is converted into a saddle-point problem. To tackle this problem, a novel double accelerated push–pull (DAPP) algorithm is proposed which concurrently employs row-stochastic (RS) and column-stochastic (CS) weight matrices. Specifically, each node pushes gradient information weighted by CS matrices to out-neighbors, while pulls dual variables fused by RS matrices from in-neighbors. Combined with CS matrices, the gradient tracking scheme is employed to promote the efficiency of tracking the averaged gradients. Furthermore, aiming at improving the convergence rate, the heavy-ball method is used to update both primal and dual variables. Then, rigorous theoretical results indicate that DAPP obtains the sublinear bounds of regret and constraint violation. Finally, the plug-in electric vehicles (PEVs) charging problem is utilized to demonstrate the validity of the proposed algorithm.

AB/Push-Pull method for distributed optimization in time-varying directed networks

In this paper, we study the distributed optimization problem for a system of agents embedded in time-varying directed communication networks. Each agent has its own cost function and agents cooperate to determine the global decision that minimizes the summation of all individual cost functions. We consider the so-called push-pull gradient-based algorithm (termed as AB/Push-Pull) which employs both row- and column-stochastic weights simultaneously to track the optimal decision and the gradient of the global cost while ensuring consensus and optimality. We show that the algorithm converges linearly to the optimal solution over a time-varying directed network for a constant stepsize when the agent's cost function is smooth and strongly convex. The linear convergence of the method has been shown in [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], where the multi-step consensus contraction parameters for row- and column-stochastic mixing matrices are not directly related to the underlying graph structure, and the explicit range for the stepsize value is not provided. With respect to [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], the novelty of this work is twofold: (1) we establish the one-step consensus contraction for both row- and column-stochastic mixing matrices with the contraction parameters given explicitly in terms of the graph diameter and other graph properties; and (2) we provide explicit upper bounds for the stepsize value in terms of the properties of the cost functions, the mixing matrices, and the graph connectivity structure.

Computable Convergence Rate Bound for Ratio Consensus Algorithms

The objective of the paper is to establish a computable upper bound for the almost sure convergence rate for a class of ratio consensus algorithms defined via column-stochastic matrices. Our result extends the works of Iutzeler et al. from 2013 on similar bounds that have been obtained in a more restrictive setup with limited conclusions. The present paper complements results by Gerencsér and Gerencsér from 2022, identifying the exact almost sure convergence rate of a wide class of ratio consensus algorithms in terms of a spectral gap, which is, however, not computable in general. The upper bound provided in the paper will be compared to the actual rate of almost sure convergence experimentally on a range of modulated random geometric graphs with random local interactions.

Β-matrices and β-tensors

In this manuscript we introduce the class of ?-matrices, which gives a new sufficient condition for the positivity of the determinant. However, we show that nonnegative ?-matrices are not necessarily P-matrices. For column stochastic matrices, the property of being a ?-matrix is weaker than strict diagonal dominance. We extend ?-matrices to tensors and call them ?-tensors. Although they are not in general P-tensors, we prove that nonnegative ?-tensors of odd order are P-tensors

Decentralized Optimization Over Time-Varying Directed Graphs With Row and Column-Stochastic Matrices

In this article, we provide a distributed optimization algorithm, termed as TV- $\mathcal {AB}$ , that minimizes a sum of convex functions over time-varying, random directed graphs. Contrary to the existing work, the algorithm we propose does not require eigenvector estimation to estimate the (non- $\mathbf {1}$ ) Perron eigenvector of a stochastic matrix. Instead, the proposed approach relies on a novel information mixing approach that exploits both row- and column-stochastic weights to achieve agreement toward the optimal solution when the underlying graph is directed. We show that TV- $\mathcal {AB}$ converges linearly to the optimal solution when the global objective is smooth and strongly convex, and the underlying time-varying graphs exhibit bounded connectivity, i.e., a union of every $C$ consecutive graphs is strongly connected. We derive the convergence results based on the stability analysis of a linear system of inequalities along with a matrix perturbation argument. Simulations confirm the findings in this article.

Distributed Nonlinear Estimation Over Unbalanced Directed Networks

This paper investigates distributed nonlinear parameter estimation in unbalanced multi-agent networks, where individual agents sequentially make local, nonlinear, noisy measurements of the true but unknown parameter, and aim at estimating the parameter consistently by sharing their measurements with their neighbors via an arbitrarily predesigned directed communication topology. To this end, a distributed nonlinear least-squares estimator, termed as DNLS-AB, is proposed by using row- and column-stochastic matrices associated with the communication topology. Under the strong connectivity of the communication topology and several mild assumptions on the sensing model functions, it is shown that the DNLS-AB estimator leads to consistent parameter estimates if the stepsize is chosen small enough. Furthermore, the performance of the designed algorithm in terms of the asymptotic mean and asymptotic covariance of the error sequence between the weighted estimate sequence and the true value of the parameter is also analyzed and shown to be consistent with that of one widely used centralized estimator. Finally, a numerical example is given to validate the proposed algorithm.

Randomized Gradient-Free Distributed Optimization Methods for a Multiagent System With Unknown Cost Function

This paper proposes a randomized gradient-free distributed optimization algorithm to solve a multiagent optimization problem with set constraints. Random gradient-free oracle instead of the true gradient information is built locally such that the estimated gradient information is utilized in guiding the update of decision variables. Thus, the algorithm requires no explicit expressions but only local measurements of the cost functions. The row-stochastic and column-stochastic matrices are used as the weighting matrices during the communication with neighbors, making the algorithm convenient to implement in directed graphs as compared with the doubly stochastic weighting matrix. Without the true gradient information, we establish asymptotic convergence to the approximated optimal solution, where the optimality gap can be set arbitrarily small. Moreover, it is shown that the proposed algorithm achieves the same rate of convergence $O(\ln t/\sqrt{t})$ as the state-of-the-art gradient-based methods with similar settings, but having the advantages of less required information and more practical communication topologies.

Linear Convergence in Optimization Over Directed Graphs With Row-Stochastic Matrices

This paper considers a distributed optimization problem over a multi-agent network, in which the objective function is a sum of individual cost functions at the agents. We focus on the case when communication between the agents is described by a \emph{directed} graph. Existing distributed optimization algorithms for directed graphs require at least the knowledge of the neighbors' out-degree at each agent (due to the requirement of column-stochastic matrices). In contrast, our algorithm requires no such knowledge. Moreover, the proposed algorithm achieves the best known rate of convergence for this class of problems, $O(\mu^k)$ for $0<\mu<1$, where $k$ is the number of iterations, given that the objective functions are strongly-convex and have Lipschitz-continuous gradients. Numerical experiments are also provided to illustrate the theoretical findings.

On the distributed optimization over directed networks

In this paper, we propose a distributed algorithm, called Directed-Distributed Subgradient Descent (D-DSD), to solve multi-agent optimization problems over directed graphs. Existing algorithms mostly deal with similar problems under the assumption of undirected networks, i.e., requiring the weight matrices to be doubly-stochastic. The row-stochasticity of the weight matrix guarantees that all agents reach consensus, while the column-stochasticity ensures that each agent’s local (sub)gradient contributes equally to the global objective. In a directed graph, however, it may not be possible to construct a doubly-stochastic weight matrix in a distributed manner. We overcome this difficulty by augmenting an additional variable for each agent to record the change in the state evolution. In each iteration, the algorithm simultaneously constructs a row-stochastic matrix and a column-stochastic matrix instead of only a doubly-stochastic matrix. The convergence of the new weight matrix, depending on the row-stochastic and column-stochastic matrices, ensures agents to reach both consensus and optimality. The analysis shows that the proposed algorithm converges at a rate of O(lnkk), where k is the number of iterations.

Animal population social structure models

We develop a new class of mathematical ranking models for the purpose of quantifying the social structure in animal populations. Our approach is based on taking into account both the interaction between single individuals as well as their role within their community as a whole. From a mathematical point of view, these models are (possibly nonlinear) eigenvalue problems for column stochastic matrices. In order to provide a procedure for their computational treatment, we derive a suitable Newton iteration method on simplexes.

Achieving Geometric Convergence for Distributed Optimization Over Time-Varying Graphs

This paper considers the problem of distributed optimization over time-varying graphs. For the case of undirected graphs, we introduce a distributed algorithm, referred to as DIGing, based on a combination of a distributed inexact gradient method and a gradient tracking technique. The DIGing algorithm uses doubly stochastic mixing matrices and employs fixed step-sizes and, yet, drives all the agents' iterates to a global and consensual minimizer. When the graphs are directed, in which case the implementation of doubly stochastic mixing matrices is unrealistic, we construct an algorithm that incorporates the push-sum protocol into the DIGing structure, thus obtaining the Push-DIGing algorithm. Push-DIGing uses column stochastic matrices and fixed step-sizes, but it still converges to a global and consensual minimizer. Under the strong convexity assumption, we prove that the algorithms converge at R-linear (geometric) rates as long as the step-sizes do not exceed some upper bounds. We establish explicit est...

Inequalities for convex sequences and nondecreasing convex functions

In this paper, by using Wu–Debnath’s method, we establish inequalities of Hammer–Bullen type for convex sequences and nondecreasing convex functions. In particular, we give an extension of a recent Farissi’s inequality. By assuming the symmetry of an involved sequence, we derive an inequality of Fejer–Hammer–Bullen type for convex sequences. In addition, we establish some companions of an Hermite–Hadamard like inequality by Latreuch and Belaidi. In our approach we use matrix methods based on column stochastic matrices.

Finding the stationary states of Markov chains by iterative methods

In this paper, we develop new methods for approximating dominant eigenvector of column-stochastic matrices. We analyze the Google matrix, and present an averaging scheme with linear rate of convergence in terms of 1-norm distance. For extending this convergence result onto general case, we assume existence of a positive row in the matrix. Our new numerical scheme, the Reduced Power Method (RPM), can be seen as a proper averaging of the power iterates of a reduced stochastic matrix. We analyze also the usual Power Method (PM) and obtain convenient conditions for its linear rate of convergence with respect to 1-norm.

Average Consensus in the Presence of Delays in Directed Graph Topologies

Classical distributed algorithms for asymptotic average consensus typically assume timely and reliable exchange of information between neighboring components of a given multi-component system. These assumptions are not necessarily valid in practice due to varying delays that might affect computations at different nodes and/or transmissions at different links. In this work, we propose a protocol that overcomes this limitation and, unlike existing consensus protocols in the presence of delays, ensures asymptotic consensus to the exact average, despite the presence of arbitrary (but bounded) delays in the communication links. The protocol requires that each component has knowledge of the number of its out-neighbors (i.e., the number of components to which it can send information) and its proof of correctness relies on the weak convergence of a backward product of column stochastic matrices. The proposed algorithm is demonstrated via illustrative examples.

Almost periodicity of stochastic operators on $\ell^1(\mathbb{N})$

We characterize from a functional analytic point of view almost periodicity of operators on $\ell^1$ given by infinite column-stochastic matrices. Some of the equivalent properties occur, under the name of Foster's condition, in the theory of stochastic processes. The results are applied to flows in infinite networks.

Convergence of Subdivision Schemes Associated with Nonnegative Masks

This paper is concerned with refinement equations of the type $$ f = \sum_{\alpha\in\bbbz^s} a(\alpha) f({M\kern .12em\cdot}-\alpha), $$ where f is the unknown function defined on the s-dimensional Euclidean space $\bbbr^s$, a is a finitely supported sequence on $\bbbz^s$, and M is an s x s dilation matrix with m := |det M|. The solution of a refinement equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach.

Sets of matrices all infinite products of which converge

An infinite product ∏ ∞ i=1 M i of matrices converges (on the right) if lim i→∞ M 1 … M i exists. A set ∑={ A i : i⩾ 1}of n x n matrices is called an RCP set (right- convergent product set) if all infinite products with each element drawn from ∑ converge. Such sets of matrices arise in constructing self-similar objects like von Koch's snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set ∑ to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in ∑ and finite products of these matrices. Necessary and sufficient conditions are given for a finite set ∑ to be an RCP set having a limit function M ∑( d)=π ∞ i=1 A d i , where d=( d 1,…, d n,…), which is a continuous function on the space of all sequences d with the sequence topology. Finite RCP sets of column-stochastic matrices are completely characterized. Some results are given on the problem of algorithmically deciding if a given set ∑ is an RCP set.

Aggregation of Markov processes: Axiomatization

We give two simple axioms that characterize a simple functional form for aggregation of column stochastic matrices (i.e., Markov processes). Several additional observations are made about such aggregation, including the special case in which the aggregated process is Markovian relative to the original one.

A geometric treatment of generalized inverses and semigroups of nonnegative matrices

The purpose of this paper is to provide a unified treatment from the geometric viewpoint of the following closely related aspects of nonnegative matrices: nonnegative matrices with nonnegative generalized inverses of various kinds; nonnegative rank factorization; regular elements, Green's relations, and maximal subgroups of the semigroups of nonnegative matrices, stochastic matrices, column stochastic matrices, and doubly stochastic matrices.