13,402 research outputs found
Diffusion Adaptation Strategies for Distributed Optimization and Learning over Networks
We propose an adaptive diffusion mechanism to optimize a global cost function
in a distributed manner over a network of nodes. The cost function is assumed
to consist of a collection of individual components. Diffusion adaptation
allows the nodes to cooperate and diffuse information in real-time; it also
helps alleviate the effects of stochastic gradient noise and measurement noise
through a continuous learning process. We analyze the mean-square-error
performance of the algorithm in some detail, including its transient and
steady-state behavior. We also apply the diffusion algorithm to two problems:
distributed estimation with sparse parameters and distributed localization.
Compared to well-studied incremental methods, diffusion methods do not require
the use of a cyclic path over the nodes and are robust to node and link
failure. Diffusion methods also endow networks with adaptation abilities that
enable the individual nodes to continue learning even when the cost function
changes with time. Examples involving such dynamic cost functions with moving
targets are common in the context of biological networks.Comment: 34 pages, 6 figures, to appear in IEEE Transactions on Signal
Processing, 201
Flow graphs: interweaving dynamics and structure
The behavior of complex systems is determined not only by the topological
organization of their interconnections but also by the dynamical processes
taking place among their constituents. A faithful modeling of the dynamics is
essential because different dynamical processes may be affected very
differently by network topology. A full characterization of such systems thus
requires a formalization that encompasses both aspects simultaneously, rather
than relying only on the topological adjacency matrix. To achieve this, we
introduce the concept of flow graphs, namely weighted networks where dynamical
flows are embedded into the link weights. Flow graphs provide an integrated
representation of the structure and dynamics of the system, which can then be
analyzed with standard tools from network theory. Conversely, a structural
network feature of our choice can also be used as the basis for the
construction of a flow graph that will then encompass a dynamics biased by such
a feature. We illustrate the ideas by focusing on the mathematical properties
of generic linear processes on complex networks that can be represented as
biased random walks and also explore their dual consensus dynamics.Comment: 4 pages, 1 figur
An analytical framework for a consensus-based global optimization method
In this paper we provide an analytical framework for investigating the
efficiency of a consensus-based model for tackling global optimization
problems. This work justifies the optimization algorithm in the mean-field
sense showing the convergence to the global minimizer for a large class of
functions. Theoretical results on consensus estimates are then illustrated by
numerical simulations where variants of the method including nonlinear
diffusion are introduced
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Diffusion-Based Adaptive Distributed Detection: Steady-State Performance in the Slow Adaptation Regime
This work examines the close interplay between cooperation and adaptation for
distributed detection schemes over fully decentralized networks. The combined
attributes of cooperation and adaptation are necessary to enable networks of
detectors to continually learn from streaming data and to continually track
drifts in the state of nature when deciding in favor of one hypothesis or
another. The results in the paper establish a fundamental scaling law for the
steady-state probabilities of miss-detection and false-alarm in the slow
adaptation regime, when the agents interact with each other according to
distributed strategies that employ small constant step-sizes. The latter are
critical to enable continuous adaptation and learning. The work establishes
three key results. First, it is shown that the output of the collaborative
process at each agent has a steady-state distribution. Second, it is shown that
this distribution is asymptotically Gaussian in the slow adaptation regime of
small step-sizes. And third, by carrying out a detailed large deviations
analysis, closed-form expressions are derived for the decaying rates of the
false-alarm and miss-detection probabilities. Interesting insights are gained.
In particular, it is verified that as the step-size decreases, the error
probabilities are driven to zero exponentially fast as functions of ,
and that the error exponents increase linearly in the number of agents. It is
also verified that the scaling laws governing errors of detection and errors of
estimation over networks behave very differently, with the former having an
exponential decay proportional to , while the latter scales linearly
with decay proportional to . It is shown that the cooperative strategy
allows each agent to reach the same detection performance, in terms of
detection error exponents, of a centralized stochastic-gradient solution.Comment: The paper will appear in IEEE Trans. Inf. Theor
A Multitask Diffusion Strategy with Optimized Inter-Cluster Cooperation
We consider a multitask estimation problem where nodes in a network are
divided into several connected clusters, with each cluster performing a
least-mean-squares estimation of a different random parameter vector. Inspired
by the adapt-then-combine diffusion strategy, we propose a multitask diffusion
strategy whose mean stability can be ensured whenever individual nodes are
stable in the mean, regardless of the inter-cluster cooperation weights. In
addition, the proposed strategy is able to achieve an asymptotically unbiased
estimation, when the parameters have same mean. We also develop an
inter-cluster cooperation weights selection scheme that allows each node in the
network to locally optimize its inter-cluster cooperation weights. Numerical
results demonstrate that our approach leads to a lower average steady-state
network mean-square deviation, compared with using weights selected by various
other commonly adopted methods in the literature.Comment: 30 pages, 8 figures, submitted to IEEE Journal of Selected Topics in
Signal Processin
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