508 research outputs found
Conditions for rapid mixing of parallel and simulated tempering on multimodal distributions
We give conditions under which a Markov chain constructed via parallel or
simulated tempering is guaranteed to be rapidly mixing, which are applicable to
a wide range of multimodal distributions arising in Bayesian statistical
inference and statistical mechanics. We provide lower bounds on the spectral
gaps of parallel and simulated tempering. These bounds imply a single set of
sufficient conditions for rapid mixing of both techniques. A direct consequence
of our results is rapid mixing of parallel and simulated tempering for several
normal mixture models, and for the mean-field Ising model.Comment: Published in at http://dx.doi.org/10.1214/08-AAP555 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Parallel Tempering: Theory, Applications, and New Perspectives
We review the history of the parallel tempering simulation method. From its
origins in data analysis, the parallel tempering method has become a standard
workhorse of physiochemical simulations. We discuss the theory behind the
method and its various generalizations. We mention a selected set of the many
applications that have become possible with the introduction of parallel
tempering and we suggest several promising avenues for future research.Comment: 21 pages, 4 figure
Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs
The effectiveness of stochastic algorithms based on Monte Carlo dynamics in
solving hard optimization problems is mostly unknown. Beyond the basic
statement that at a dynamical phase transition the ergodicity breaks and a
Monte Carlo dynamics cannot sample correctly the probability distribution in
times linear in the system size, there are almost no predictions nor intuitions
on the behavior of this class of stochastic dynamics. The situation is
particularly intricate because, when using a Monte Carlo based algorithm as an
optimization algorithm, one is usually interested in the out of equilibrium
behavior which is very hard to analyse. Here we focus on the use of Parallel
Tempering in the search for the largest independent set in a sparse random
graph, showing that it can find solutions well beyond the dynamical threshold.
Comparison with state-of-the-art message passing algorithms reveals that
parallel tempering is definitely the algorithm performing best, although a
theory explaining its behavior is still lacking.Comment: 14 pages, 12 figure
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