Quantitative bounds on convergence of time-inhomogeneous Markov chains

Convergence rates of Markov chains have been widely studied in recent years. In particular, quantitative bounds on convergence rates have been studied in various forms by Meyn and Tweedie [Ann. Appl. Probab. 4 (1994) 981-1101], Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566], Roberts and Tweedie [Stochastic Process. Appl. 80 (1999) 211-229], Jones and Hobert [Statist. Sci. 16 (2001) 312-334] and Fort [Ph.D. thesis (2001) Univ. Paris VI]. In this paper, we extend a result of Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566] that concerns quantitative convergence rates for time-homogeneous Markov chains. Our extension allows us to consider f-total variation distance (instead of total variation) and time-inhomogeneous Markov chains. We apply our results to simulated annealing.Comment: Published at http://dx.doi.org/10.1214/105051604000000620 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Convergence of adaptive and interacting Markov chain Monte Carlo algorithms

Adaptive and interacting Markov chain Monte Carlo algorithms (MCMC) have been recently introduced in the literature. These novel simulation algorithms are designed to increase the simulation efficiency to sample complex distributions. Motivated by some recently introduced algorithms (such as the adaptive Metropolis algorithm and the interacting tempering algorithm), we develop a general methodological and theoretical framework to establish both the convergence of the marginal distribution and a strong law of large numbers. This framework weakens the conditions introduced in the pioneering paper by Roberts and Rosenthal [J. Appl. Probab. 44 (2007) 458--475]. It also covers the case when the target distribution $\pi$ is sampled by using Markov transition kernels with a stationary distribution that differs from $\pi$.Comment: Published in at http://dx.doi.org/10.1214/11-AOS938 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Polynomial ergodicity of Markov transition kernels

AbstractThis paper discusses quantitative bounds on the convergence rates of Markov chains, under conditions implying polynomial convergence rates. This paper extends an earlier work by Roberts and Tweedie (Stochastic Process. Appl. 80(2) (1999) 211), which provides quantitative bounds for the total variation norm under conditions implying geometric ergodicity.Explicit bounds for the total variation norm are obtained by evaluating the moments of an appropriately defined coupling time, using a set of drift conditions, adapted from an earlier work by Tuominen and Tweedie (Adv. Appl. Probab. 26(3) (1994) 775). Applications of this result are then presented to study the convergence of random walk Hastings Metropolis algorithm for super-exponential target functions and of general state-space models. Explicit bounds for f-ergodicity are also given, for an appropriately defined control function f

Application of semi-definite relaxation to multiuser detection in a CDMA context

- De nombreuses problématiques de traitement du signal se ramènent à la résolution d'un problème d'optimisation combinatoire. Récemment, la Relaxation Semi-Définie (SDR) s'est révélée être une approche prometteuse en la matière, permettant une relaxation réaliste de problèmes NP-complets. Dans cet article, nous présentons un algorithme efficace pour résoudre SDR avec une complexité réduite. L'objet principale est d'étudier des méthodes de programmation non linéaires qui reposent sur un changement de variable consistant à remplacer la variable symétrique définie positive X intervenant dans SDR par une variable rectangulaire V à travers la décomposition X =V TV. Des résultats récents sur les rangs de matrices de corrélations extrémales permettent de conduire à un algorithme de faible complexité avec une perte négligeable en matière de performances. Des résultats très encourageants ont été obtenus pour résoudre des problèmes d'optimisation combinatoire de grande dimension, tel que celui qui intervient dans la détection multi utilisateur en mode CDMA

Blocking strategies and stability of particle Gibbs samplers

Sampling from the posterior probability distribution of the latent states of a hidden Markov model is non-trivial even in the context of Markov chain Monte Carlo. To address this Andrieu et al. (2010) proposed a way of using a particle filter to construct a Markov kernel that leaves his posterior distribution invariant. Recent theoretical results establish the uniform ergodicity of this Markov kernel and show that the mixing rate does not deteriorate provided the number of particles grows at least linearly with the number of latent states. However, this gives rise to a cost per application of the kernel that is quadratic in the number of latent states, which can be prohibitive for long observation sequences. Using blocking strategies, we devise samplers that have a stable mixing rate for a cost per iteration that is linear in the number of latent states and which are easily parallelizable.The authors thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Monte Carlo Inference for Complex Statistical Models when work on this paper was undertaken. This work was supported by the Engineering and Physical Sciences Research Council [grant numbers EP/K020153/1, EP/K032208/1] and the Swedish Research Council [contract number 2016-04278]

Estimation robuste de l'enveloppe spectrale d'un signal harmonique bruité

Les modèles sinusoïdaux de parole nécessitent l'estimation d'une enveloppe spectrale reliant les différents harmoniques. Une nouvelle méthode est développée ici ; elle repose sur un critère de vraisemblance pénalisée obtenu à partir du comportement statistique des carrés des amplitudes des sinusoïdes, estimés en présence de bruit de mesure. Un critère composite est développé afin de traiter les trames mixtes, dans lesquelles le signal utile de parole est composé de sinusoïdes et de bruit

Time series prediction via aggregation : an oracle bound including numerical cost

We address the problem of forecasting a time series meeting the Causal Bernoulli Shift model, using a parametric set of predictors. The aggregation technique provides a predictor with well established and quite satisfying theoretical properties expressed by an oracle inequality for the prediction risk. The numerical computation of the aggregated predictor usually relies on a Markov chain Monte Carlo method whose convergence should be evaluated. In particular, it is crucial to bound the number of simulations needed to achieve a numerical precision of the same order as the prediction risk. In this direction we present a fairly general result which can be seen as an oracle inequality including the numerical cost of the predictor computation. The numerical cost appears by letting the oracle inequality depend on the number of simulations required in the Monte Carlo approximation. Some numerical experiments are then carried out to support our findings

On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case

We study the problem of sampling from a probability distribution π on Rd which has a density w.r.t. the Lebesgue measure known up to a normalization factor x→ e−U(x)/f Rd e−U(y) dy. We analyze a sampling method based on the Euler discretization of the Langevin stochastic differential equations under the assumptions that the potential U is continuously differentiable, ∇U is Lipschitz, and U is strongly concave. We focus on the case where the gradient of the log-density cannot be directly computed but unbiased estimates of the gradient from possibly dependent observations are available. This setting can be seen as a combination of a stochastic approximation (here stochastic gradient) type algorithms with discretized Langevin dynamics. We obtain an upper bound of the Wasserstein-2 distance between the law of the iterates of this algorithm and the target distribution π with constants depending explicitly on the Lipschitz and strong convexity constants of the potential and the dimension of the space. Finally, under weaker assumptions on U and its gradient but in the presence of independent observations, we obtain analogous results in Wasserstein-2 distance. © 2021 ISI/B