12 research outputs found
On the Connectivity of Unions of Random Graphs
Graph-theoretic tools and techniques have seen wide use in the multi-agent
systems literature, and the unpredictable nature of some multi-agent
communications has been successfully modeled using random communication graphs.
Across both network control and network optimization, a common assumption is
that the union of agents' communication graphs is connected across any finite
interval of some prescribed length, and some convergence results explicitly
depend upon this length. Despite the prevalence of this assumption and the
prevalence of random graphs in studying multi-agent systems, to the best of our
knowledge, there has not been a study dedicated to determining how many random
graphs must be in a union before it is connected. To address this point, this
paper solves two related problems. The first bounds the number of random graphs
required in a union before its expected algebraic connectivity exceeds the
minimum needed for connectedness. The second bounds the probability that a
union of random graphs is connected. The random graph model used is the
Erd\H{o}s-R\'enyi model, and, in solving these problems, we also bound the
expectation and variance of the algebraic connectivity of unions of such
graphs. Numerical results for several use cases are given to supplement the
theoretical developments made.Comment: 16 pages, 3 tables; accepted to 2017 IEEE Conference on Decision and
Control (CDC
Distributed Stochastic Optimization over Time-Varying Noisy Network
This paper is concerned with distributed stochastic multi-agent optimization
problem over a class of time-varying network with slowly decreasing
communication noise effects. This paper considers the problem in composite
optimization setting which is more general in noisy network optimization. It is
noteworthy that existing methods for noisy network optimization are Euclidean
projection based. We present two related different classes of non-Euclidean
methods and investigate their convergence behavior. One is distributed
stochastic composite mirror descent type method (DSCMD-N) which provides a more
general algorithm framework than former works in this literature. As a
counterpart, we also consider a composite dual averaging type method (DSCDA-N)
for noisy network optimization. Some main error bounds for DSCMD-N and DSCDA-N
are obtained. The trade-off among stepsizes, noise decreasing rates,
convergence rates of algorithm is analyzed in detail. To the best of our
knowledge, this is the first work to analyze and derive convergence rates of
optimization algorithm in noisy network optimization. We show that an optimal
rate of in nonsmooth convex optimization can be obtained for
proposed methods under appropriate communication noise condition. Moveover,
convergence rates in different orders are comprehensively derived in both
expectation convergence and high probability convergence sense.Comment: 27 page
Robust Distributed Averaging on Networks with Adversarial Intervention
We study the interaction between a network designer and an adversary over a
dynamical network. The network consists of nodes performing continuous-time
distributed averaging. The goal of the network designer is to assist the nodes
reach consensus by changing the weights of a limited number of links in the
network. Meanwhile, an adversary strategically disconnects a set of links to
prevent the nodes from converging. We formulate two problems to describe this
competition where the order in which the players act is reversed in the two
problems. We utilize Pontryagin's Maximum Principle (MP) to tackle both
problems and derive the optimal strategies. Although the canonical equations
provided by the MP are intractable, we provide an alternative characterization
for the optimal strategies that highlights a connection with potential theory.
Finally, we provide a sufficient condition for the existence of a saddle-point
equilibrium (SPE) for this zero-sum game.Comment: 8 pages, 2 figures, submitted to CDC 201