Fully Asynchronous Push-Sum With Growing Intercommunication Intervals

Abstract

We propose an algorithm for average consensus over a directed graph which is both fully asynchronous and robust to unreliable communications. We show its convergence to the average, while allowing for slowly growing but potentially unbounded communication failures. I. INTRODUCTIONConsider a set of agents, whose goal is to reach consensus by exchanging information locally with their neighbors through a directed graph. There is a large body of work on consensus algorithms. Ordinary consensus has been shown to converge asymptotically under various scenarios such as growing intercommunicating intervals [1], presence of delays and/or unbounded intercommunication intervals [2]. Another problem of interest for which extensive research has been carried out is average consensus. While most related works study asymptotic convergence, [3] studies average consensus in a finite number of steps. Push-sum is one of the many algorithms for average consensus that was first proposed by [4]. This algorithm has been widely used to develop protocols that reach average consensus, under different assumptions and scenarios; such as the presence of bounded delays [5], time varying graphs [6][7], or asynchronous communication [8].Since reliable communication is a very restrictive assumption in network applications, or expensive to enforce, recent work has considered algorithms that reach consensus in a setting where communication between agents is unreliable. While in this case, push-sum might not converge to average, exponential convergence still holds and the error between the final value and the true average can be characterized [9]. In [10], Vaidya et al. introduce the technique of running sums (counters) and modify push-sum to overcome possible packet drops and imprecise knowledge of the network in a synchronous communication setting. They prove almost surely convergence of their algorithms using weak ergodicity. Inspired by [10], [11] takes this further and develops an asynchronous algorithm for average consensus, which is robust to unreliable communication. This algorithm uses a broadcast asymmetric communication protocol; that is, at each iteration only one node is allowed to wake up and transmit information to its neighbors. Exponential convergence of this algorithm is proved under bounded consecutive link failures and nodes' update delays.Consensus and average consensus have a lot of application in other algorithms as well; they can be used as a building block to develop distributed optimization algorithms [12] [13]. For example, in [14] the authors use a robust version of push-sum as a building block to develop an asynchronous Newton-based distributed optimization algorithm, robust to packet losses.A lot of available works in the literature assume bounded intercommunication intervals; which motivated us to study and explore sufficient connectivity conditions which allow intercommunication intervals to slowly grow and potentially be unbounded. We propose logarithmically growing upper bounds which guarantee convergence.

On Increasing Self-Confidence in Non-Bayesian Social Learning over Time-Varying Directed Graphs

“…which in turn makes (9) hold. If Assumption 2 does not hold, one cannot guarantee (15) holds, because in general we only have…”

“…In [12], the authors considered a fixed rate of decay of O(1/k) and showed learning is achieved for both fixed and time-varying graphs that are always connected. Other authors have considered the phenomena of asymptotic isolation of agents in a network assuming the intervals of intercommunication between them increase with time [14], [15].…”

A Communication-Efficient Algorithm for Exponentially Fast Non-Bayesian Learning in Networks

“…Given this premise, it is natural to ask: how frequently must the agents communicate to solve the desired problem? Owing to its practical relevance, the question posed above has received significant recent interest by the control system, information theory and machine learning communities in the context of a variety of problems, namely average consensus [1], optimization [2]- [4], and static parameter estimation [5]. Our goal in this paper is to extend such investigations to the problem of non-Bayesian learning in a network, also known as the distributed hypothesis testing problem [6]- [11].…”

“…Specifically, the data-aggregation step of our algorithm involves a min-protocol as opposed to the consensusbased averaging schemes intrinsic to existing linear [6], [7] and log-linear [8]- [10] learning rules. The basic strategy we employ to achieve communication-efficiency is in line with those proposed in [1], [2], [5], where inter-agent communications become progressively sparser as time evolves. In particular, the authors in [1] and [2] explore deterministic rules where the inter-communication intervals grow logarithmically and polynomially in time, respectively.…”

“…The basic strategy we employ to achieve communication-efficiency is in line with those proposed in [1], [2], [5], where inter-agent communications become progressively sparser as time evolves. In particular, the authors in [1] and [2] explore deterministic rules where the inter-communication intervals grow logarithmically and polynomially in time, respectively. In contrast, the authors in [5] propose a rule where at each time-step, an agent communicates with its neighbors in the graph with a probability that decays to zero at a sub-linear rate.…”