287 research outputs found
Byzantine Agreement with Optimal Early Stopping, Optimal Resilience and Polynomial Complexity
We provide the first protocol that solves Byzantine agreement with optimal
early stopping ( rounds) and optimal resilience () using
polynomial message size and computation.
All previous approaches obtained sub-optimal results and used resolve rules
that looked only at the immediate children in the EIG (\emph{Exponential
Information Gathering}) tree. At the heart of our solution are new resolve
rules that look at multiple layers of the EIG tree.Comment: full version of STOC 2015 abstrac
Polynomial Linear Programming with Gaussian Belief Propagation
Interior-point methods are state-of-the-art algorithms for solving linear
programming (LP) problems with polynomial complexity. Specifically, the
Karmarkar algorithm typically solves LP problems in time O(n^{3.5}), where
is the number of unknown variables. Karmarkar's celebrated algorithm is known
to be an instance of the log-barrier method using the Newton iteration. The
main computational overhead of this method is in inverting the Hessian matrix
of the Newton iteration. In this contribution, we propose the application of
the Gaussian belief propagation (GaBP) algorithm as part of an efficient and
distributed LP solver that exploits the sparse and symmetric structure of the
Hessian matrix and avoids the need for direct matrix inversion. This approach
shifts the computation from realm of linear algebra to that of probabilistic
inference on graphical models, thus applying GaBP as an efficient inference
engine. Our construction is general and can be used for any interior-point
algorithm which uses the Newton method, including non-linear program solvers.Comment: 7 pages, 1 figure, appeared in the 46th Annual Allerton Conference on
Communication, Control and Computing, Allerton House, Illinois, Sept. 200
A Low Density Lattice Decoder via Non-Parametric Belief Propagation
The recent work of Sommer, Feder and Shalvi presented a new family of codes
called low density lattice codes (LDLC) that can be decoded efficiently and
approach the capacity of the AWGN channel. A linear time iterative decoding
scheme which is based on a message-passing formulation on a factor graph is
given.
In the current work we report our theoretical findings regarding the relation
between the LDLC decoder and belief propagation. We show that the LDLC decoder
is an instance of non-parametric belief propagation and further connect it to
the Gaussian belief propagation algorithm. Our new results enable borrowing
knowledge from the non-parametric and Gaussian belief propagation domains into
the LDLC domain. Specifically, we give more general convergence conditions for
convergence of the LDLC decoder (under the same assumptions of the original
LDLC convergence analysis). We discuss how to extend the LDLC decoder from
Latin square to full rank, non-square matrices. We propose an efficient
construction of sparse generator matrix and its matching decoder. We report
preliminary experimental results which show our decoder has comparable symbol
to error rate compared to the original LDLC decoder.%Comment: Submitted for publicatio
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