159 research outputs found
Convergence of a Particle-based Approximation of the Block Online Expectation Maximization Algorithm
Online variants of the Expectation Maximization (EM) algorithm have recently
been proposed to perform parameter inference with large data sets or data
streams, in independent latent models and in hidden Markov models.
Nevertheless, the convergence properties of these algorithms remain an open
problem at least in the hidden Markov case. This contribution deals with a new
online EM algorithm which updates the parameter at some deterministic times.
Some convergence results have been derived even in general latent models such
as hidden Markov models. These properties rely on the assumption that some
intermediate quantities are available in closed form or can be approximated by
Monte Carlo methods when the Monte Carlo error vanishes rapidly enough. In this
paper, we propose an algorithm which approximates these quantities using
Sequential Monte Carlo methods. The convergence of this algorithm and of an
averaged version is established and their performance is illustrated through
Monte Carlo experiments
Limit theorems for some adaptive MCMC algorithms with subgeometric kernels: Part II
We prove a central limit theorem for a general class of adaptive Markov Chain
Monte Carlo algorithms driven by sub-geometrically ergodic Markov kernels. We
discuss in detail the special case of stochastic approximation. We use the
result to analyze the asymptotic behavior of an adaptive version of the
Metropolis Adjusted Langevin algorithm with a heavy tailed target density.Comment: 34 page
Adaptive Equi-Energy Sampler : Convergence and Illustration
Markov chain Monte Carlo (MCMC) methods allow to sample a distribution known
up to a multiplicative constant. Classical MCMC samplers are known to have very
poor mixing properties when sampling multimodal distributions. The Equi-Energy
sampler is an interacting MCMC sampler proposed by Kou, Zhou and Wong in 2006
to sample difficult multimodal distributions. This algorithm runs several
chains at different temperatures in parallel, and allow lower-tempered chains
to jump to a state from a higher-tempered chain having an energy 'close' to
that of the current state. A major drawback of this algorithm is that it
depends on many design parameters and thus, requires a significant effort to
tune these parameters. In this paper, we introduce an Adaptive Equi-Energy
(AEE) sampler which automates the choice of the selection mecanism when jumping
onto a state of the higher-temperature chain. We prove the ergodicity and a
strong law of large numbers for AEE, and for the original Equi-Energy sampler
as well. Finally, we apply our algorithm to motif sampling in DNA sequences
Quantitative convergence rates for sub-geometric Markov chains
We provide explicit expressions for the constants involved in the
characterisation of ergodicity of sub-geometric Markov chains. The constants
are determined in terms of those appearing in the assumed drift and one-step
minorisation conditions. The result is fundamental for the study of some
algorithms where uniform bounds for these constants are needed for a family of
Markov kernels. Our result accommodates also some classes of inhomogeneous
chains.Comment: 14 page
Performance of a Distributed Stochastic Approximation Algorithm
In this paper, a distributed stochastic approximation algorithm is studied.
Applications of such algorithms include decentralized estimation, optimization,
control or computing. The algorithm consists in two steps: a local step, where
each node in a network updates a local estimate using a stochastic
approximation algorithm with decreasing step size, and a gossip step, where a
node computes a local weighted average between its estimates and those of its
neighbors. Convergence of the estimates toward a consensus is established under
weak assumptions. The approach relies on two main ingredients: the existence of
a Lyapunov function for the mean field in the agreement subspace, and a
contraction property of the random matrices of weights in the subspace
orthogonal to the agreement subspace. A second order analysis of the algorithm
is also performed under the form of a Central Limit Theorem. The
Polyak-averaged version of the algorithm is also considered.Comment: IEEE Transactions on Information Theory 201
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