Nonasymptotic convergence rates for cooperative learning over time-varying directed graphs

Abstract

We study the problem of distributed hypothesis testing with a network of agents where some agents repeatedly gain access to information about the correct hypothesis. The group objective is to globally agree on a joint hypothesis that best describes the observed data at all the nodes. We assume that the agents can interact with their neighbors in an unknown sequence of time-varying directed graphs.Following the pioneering work of Jadbabaie, Molavi, Sandroni, and Tahbaz-Salehi, we propose local learning dynamics which combine Bayesian updates at each node with a local aggregation rule of private agent signals. We show that these learning dynamics drive all agents to the set of hypotheses which best explain the data collected at all nodes as long as the sequence of interconnection graphs is uniformly strongly connected. Our main result establishes a non-asymptotic, explicit, geometric convergence rate for the learning dynamic.

Social learning and distributed hypothesis testing

“…This implies that under Assumption 4 Theorem 3 provides a tighter asymptotic rate than that in [32] and [31]. Hence, Theorem 3 strengthens Theorem 2 by extending the large deviation to larger class of distributions and by capturing the complete effect of nodes' influence in the network and the local observation statistics.…”

Distributed estimation and learning over heterogeneous networks

“…Geometric averaging and logarithmic opinion pools have a long history in Bayesian analysis and behavioral decision models [40], [41] and they can be also justified under specific behavioral assumptions [42]. The are also quite popular as a non-Bayesian update rule in distributed detection and estimation litrature [43], [44], [10], [14], [45]. In [45] the authors use a logarithmic opinion pool to combine the estimated posterior probability distributions in a Bayesian consensus filter; and show that as a result: the sum of KullbackLeibler divergences between the consensual probability distribution and the local posterior probability distributions is minimized.…”

“…≤ λ 1 (I + A) = 2. Susequently, we can employ the eigendecomposition of (I + A) to analyze the behavior of (I + A) t+1 in (10). Specifically, we can take a set of biorthonormal vectors l i , r i as the left and right eigenvectors corresponding to the ith eigenvalue of I +A, satisfying: l i 2 = r i 2 = 1, l T i r i = 1 for all i and l T i r j = 0, i = j; in particular, l 1 = r 1 = (1/ √ n)1.…”

“…There is a large body of literature on decentralized detection with the notable examples of [1], [2], [3]; and recently there is a renewed interest in this topic due to its applications to sensor and robotic networks [4], [5], [6], [7], [8] and the emergence of new literature considering network of sensor and computational units [9], [10], [11]. Other relevant results investigate the formation and evolution of beliefs in social networks and subsequent shaping of the individual and mass behavior through social learning [12], [13], [14].…”

“…Furthermore, we can use (11) and (10) to bound the distance between φ i,t (θ,θ) and (2 t /n)Λ(θ,θ) for any i, as follows:…”