This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations

/pdf/consensus-and-cooperation-in-networked-multi-agent-systems-5ame7z72ny.pdf

Consensus and Cooperation in Networked Multi-Agent Systems

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained l1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms l1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted l1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the l1 norm of the coefficient sequence as is common, but by reweighting the l1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing.

/pdf/enhancing-sparsity-by-reweighted-l-1-minimization-znwoz4koxr.pdf

Enhancing Sparsity by Reweighted ℓ 1 Minimization

The term Federated Learning was coined as recently as 2016 to describe a machine learning setting where multiple entities collaborate in solving a machine learning problem, under the coordination of a central server or service provider. Each client’s raw data is stored locally and not exchanged or transferred; instead, focused updates intended for immediate aggregation are used to achieve the learning objective. Since then, the topic has gathered much interest across many different disciplines and the realization that solving many of these interdisciplinary problems likely requires not just machine learning but techniques from distributed optimization, cryptography, security, differential privacy, fairness, compressed sensing, systems, information theory, statistics, and more. This monograph has contributions from leading experts across the disciplines, who describe the latest state-of-the art from their perspective. These contributions have been carefully curated into a comprehensive treatment that enables the reader to understand the work that has been done and get pointers to where effort is required to solve many of the problems before Federated Learning can become a reality in practical systems. Researchers working in the area of distributed systems will find this monograph an enlightening read that may inspire them to work on the many challenging issues that are outlined. This monograph will get the reader up to speed quickly and easily on what is likely to become an increasingly important topic: Federated Learning.

/pdf/advances-and-open-problems-in-federated-learning-3t0jgb90ia.pdf

Advances and open problems in federated learning

We study a distributed computation model for optimizing a sum of convex objective functions corresponding to multiple agents. For solving this (not necessarily smooth) optimization problem, we consider a subgradient method that is distributed among the agents. The method involves every agent minimizing his/her own objective function while exchanging information locally with other agents in the network over a time-varying topology. We provide convergence results and convergence rate estimates for the subgradient method. Our convergence rate results explicitly characterize the tradeoff between a desired accuracy of the generated approximate optimal solutions and the number of iterations needed to achieve the accuracy.

http://www.ifp.illinois.edu/~angelia/distributed_journal_final.pdf

Distributed Subgradient Methods for Multi-Agent Optimization

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained L1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms L1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted L1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations--not by reweighting the L1 norm of the coefficient sequence as is common, but by reweighting the L1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as compressed sensing.

Enhancing Sparsity by Reweighted L1 Minimization

Motivated by applications to sensor, peer-to-peer, and ad hoc networks, we study distributed algorithms, also known as gossip algorithms, for exchanging information and for computing in an arbitrarily connected network of nodes. The topology of such networks changes continuously as new nodes join and old nodes leave the network. Algorithms for such networks need to be robust against changes in topology. Additionally, nodes in sensor networks operate under limited computational, communication, and energy resources. These constraints have motivated the design of "gossip" algorithms: schemes which distribute the computational burden and in which a node communicates with a randomly chosen neighbor. We analyze the averaging problem under the gossip constraint for an arbitrary network graph, and find that the averaging time of a gossip algorithm depends on the second largest eigenvalue of a doubly stochastic matrix characterizing the algorithm. Designing the fastest gossip algorithm corresponds to minimizing this eigenvalue, which is a semidefinite program (SDP). In general, SDPs cannot be solved in a distributed fashion; however, exploiting problem structure, we propose a distributed subgradient method that solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities derived from the gossip algorithm. We use this connection to study the performance and scaling of gossip algorithms on two popular networks: Wireless Sensor Networks, which are modeled as Geometric Random Graphs, and the Internet graph under the so-called Preferential Connectivity (PC) model.

/pdf/randomized-gossip-algorithms-2g2zm9depe.pdf

Randomized gossip algorithms

Motivated by applications to sensor, peer-to-peer and ad hoc networks, we study distributed asynchronous algorithms, also known as gossip algorithms, for computation and information exchange in an arbitrarily connected network of nodes. Nodes in such networks operate under limited computational, communication and energy resources. These constraints naturally give rise to "gossip" algorithms: schemes which distribute the computational burden and in which a node communicates with a randomly chosen neighbor. We analyze the averaging problem under the gossip constraint for arbitrary network, and find that the averaging time of a gossip algorithm depends on the second largest eigenvalue of a doubly stochastic matrix characterizing the algorithm. Using recent results of Boyd, Diaconis and Xiao (2003), we show that minimizing this quantity to design the fastest averaging algorithm on the network is a semi-definite program (SDP). In general, SDPs cannot be solved distributedly; however, exploiting problem structure, we propose a subgradient method that distributedly solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities that are derived from the gossip algorithm. We use this connection to study the performance of gossip algorithm on two popular networks: wireless sensor networks, which are modeled as geometric random graphs, and the Internet graph under the so-called preferential connectivity model.

/pdf/gossip-algorithms-design-analysis-and-applications-4x6ypmmboc.pdf

Gossip algorithms: design, analysis and applications

The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network analysis, including, for example, Markov chains and continuous-time averaging networks. In this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance, i.e., the sum of the resistances between all pairs of nodes. We show that this is a convex optimization problem and can be solved efficiently either numerically or, in some cases, analytically. We show that optimal allocation of the edge weights can reduce the total effective resistance of the graph (compared to uniform weights) by a factor that grows unboundedly with the size of the graph. We show that among all graphs with $n$ nodes, the path has the largest value of optimal total effective resistance and the complete graph has the least.

/pdf/minimizing-effective-resistance-of-a-graph-1rz525m2td.pdf

Minimizing Effective Resistance of a Graph

The algebraic connectivity of a graph is the second smallest eigenvalue of the graph Laplacian, and is a measure of how well-connected the graph is. We study the problem of adding edges (from a set of candidate edges) to a graph so as to maximize its algebraic connectivity. This is a difficult combinatorial optimization, so we seek a heuristic for approximately solving the problem. The standard convex relaxation of the problem can be expressed as a semidefinite program (SDP); for modest sized problems, this yields a cheaply computable upper bound on the optimal value, as well as a heuristic for choosing the edges to be added. We describe a new greedy heuristic for the problem. The heuristic is based on the Fiedler vector, and therefore can be applied to very large graphs.

/pdf/growing-well-connected-graphs-1xhqd620j4.pdf

Growing Well-connected Graphs

A geometric random graph, Gd(n, r), is formed as follows: place n nodes uniformly at random onto the surface of the d-dimensional unit torus and connect nodes which are within a distance r of each other The Gd(n, r) has been of great interest due to its success as a model for ad-hoc wireless networks It is well known that the connectivity of Gd(n, r) exhibits a threshold property: there exists a constant αd such that for any " > 0, for rd αd(1 + ") log n/n the Gd(n, r) is connected whp In this paper, we study mixing properties of random walks on Gd(n, r) for rd(n) = ω(log n/n) Specifically, we study the scaling of mixing times of the fastest-mixing reversible random walk, and the natural random walk We find that the mixing time of both of these random walks have the same scaling laws and scale proportional to r−2 (for all d) These results hold for Gd(n, r) when distance is defined using any Lp norm Though the results of this paper are not so surprising, they are nontrivial and require new methods To obtain the scaling law for the fastest-mixing reversible random walk, we first explicitly characterize the fastest-mixing reversible random walk on a regular (grid-type) graph in d dimensions We subsequently use this to bound the mixing time of the fastest-mixing random walk on Gd(n, r) In the course of our analysis, we obtain a tight relation between the mixing time of the fastest-mixing symmetric random walk and the fastest-mixing reversible random walk with a specified equilibrium distribution on an arbitrary graph ∗Author names appear in alphabetical order †This work is supported in part by a Stanford Graduate Fellowship, and by C2S2, the MARCO Focus Center for Circuit and System Solution, under MARCO contract 2003-CT-888 1In this paper, with high probability (whp) means with probability at least 1 − 1/n2 To study the natural random walk, we first generalize a method of [DS91] to bound eigenvalues based on Poincare’s inequality and then apply it to the Gd(n, r) graph We note that the methods utilized in this paper are novel and general enough to be useful in the context of other graphs

/pdf/mixing-times-for-random-walks-on-geometric-random-graphs-45ew5go4a2.pdf

Mixing Times for Random Walks on Geometric Random Graphs.

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