17,433 research outputs found
Local stability of Belief Propagation algorithm with multiple fixed points
A number of problems in statistical physics and computer science can be
expressed as the computation of marginal probabilities over a Markov random
field. Belief propagation, an iterative message-passing algorithm, computes
exactly such marginals when the underlying graph is a tree. But it has gained
its popularity as an efficient way to approximate them in the more general
case, even if it can exhibits multiple fixed points and is not guaranteed to
converge. In this paper, we express a new sufficient condition for local
stability of a belief propagation fixed point in terms of the graph structure
and the beliefs values at the fixed point. This gives credence to the usual
understanding that Belief Propagation performs better on sparse graphs.Comment: arXiv admin note: substantial text overlap with arXiv:1101.417
The Role of Normalization in the Belief Propagation Algorithm
An important part of problems in statistical physics and computer science can
be expressed as the computation of marginal probabilities over a Markov Random
Field. The belief propagation algorithm, which is an exact procedure to compute
these marginals when the underlying graph is a tree, has gained its popularity
as an efficient way to approximate them in the more general case. In this
paper, we focus on an aspect of the algorithm that did not get that much
attention in the literature, which is the effect of the normalization of the
messages. We show in particular that, for a large class of normalization
strategies, it is possible to focus only on belief convergence. Following this,
we express the necessary and sufficient conditions for local stability of a
fixed point in terms of the graph structure and the beliefs values at the fixed
point. We also explicit some connexion between the normalization constants and
the underlying Bethe Free Energy
An FPTAS for Bargaining Networks with Unequal Bargaining Powers
Bargaining networks model social or economic situations in which agents seek
to form the most lucrative partnership with another agent from among several
alternatives. There has been a flurry of recent research studying Nash
bargaining solutions (also called 'balanced outcomes') in bargaining networks,
so that we now know when such solutions exist, and also that they can be
computed efficiently, even by market agents behaving in a natural manner. In
this work we study a generalization of Nash bargaining, that models the
possibility of unequal 'bargaining powers'. This generalization was introduced
in [KB+10], where it was shown that the corresponding 'unequal division' (UD)
solutions exist if and only if Nash bargaining solutions exist, and also that a
certain local dynamics converges to UD solutions when they exist. However, the
bound on convergence time obtained for that dynamics was exponential in network
size for the unequal division case. This bound is tight, in the sense that
there exists instances on which the dynamics of [KB+10] converges only after
exponential time. Other approaches, such as the one of Kleinberg and Tardos, do
not generalize to the unsymmetrical case. Thus, the question of computational
tractability of UD solutions has remained open. In this paper, we provide an
FPTAS for the computation of UD solutions, when such solutions exist. On a
graph G=(V,E) with weights (i.e. pairwise profit opportunities) uniformly
bounded above by 1, our FPTAS finds an \eps-UD solution in time
poly(|V|,1/\eps). We also provide a fast local algorithm for finding \eps-UD
solution, providing further justification that a market can find such a
solution.Comment: 18 pages; Amin Saberi (Ed.): Internet and Network Economics - 6th
International Workshop, WINE 2010, Stanford, CA, USA, December 13-17, 2010.
Proceedings
- …