Optimal scaling of MaLa for nonlinear regression

Breyer, L. A.; Piccioni, Mauro; Scarlatti, Sergio

doi:10.1214/105051604000000369

Cited by 23 publications

(28 citation statements)

References 15 publications

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“…context involved looking at restricted classes of models, see e.g. [17,16,7]. The most recent contributions in this still-open research direction have looked at target distributions in high-dimensions defined as changes of measure from Gaussian laws ( [12,45,49]).…”

Section: Beyond Iid Targetsmentioning

confidence: 99%

On the stability of sequential Monte Carlo methods in high dimensions

Beskos¹,

Crisan²,

Jasra³

2014

Ann. Appl. Probab.

144

178

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We investigate the stability of a Sequential Monte Carlo (SMC) method applied to the problem of sampling from a target distribution on R d for large d. It is well known [9,14,56] that using a single importance sampling step one produces an approximation for the target that deteriorates as the dimension d increases, unless the number of Monte Carlo samples N increases at an exponential rate in d. We show that this degeneracy can be avoided by introducing a sequence of artificial targets, starting from a 'simple' density and moving to the one of interest, using an SMC method to sample from the sequence (see e.g. [20,27,38,48]). Using this class of SMC methods with a fixed number of samples, one can produce an approximation for which the effective sample size (ESS) converges to a random variable εN as d → ∞ with 1 < εN < N . The convergence is achieved with a computational cost proportional to N d 2 . If εN ≪ N , we can raise its value by introducing a number of resampling steps, say m (where m is independent of d). In this case, ESS converges to a random variable εN,m as d → ∞ and limm→∞ εN,m = N . Also, we show that the Monte Carlo error for estimating a fixed dimensional marginal expectation is of order 1 √ N uniformly in d. The results imply that, in high dimensions, SMC algorithms can efficiently control the variability of the importance sampling weights and estimate fixed dimensional marginals at a cost which is less than exponential in d and indicate that, in high dimensions, resampling leads to a reduction in the Monte Carlo error and increase in the ESS.

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Section: Beyond Iid Targetsmentioning

confidence: 99%

On the stability of sequential Monte Carlo methods in high dimensions

Beskos¹,

Crisan²,

Jasra³

2014

Ann. Appl. Probab.

144

178

View full text Add to dashboard Cite

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“…To the best of our knowledge, the only paper to consider the optimal scaling for the MALA algorithm for nonproduct targets is [9], in the context of nonlinear regression. In [9] the target measure has a structure similar to that of the mean field models studied in statistical mechanics and hence behaves asymptotically like a product measure when the dimension goes to infinity. Thus the diffusion limit obtained in [9] is finite dimensional.…”

Section: Harvard University Warwick University and Warwick Universitymentioning

confidence: 99%

Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions

Pillai¹,

Stuart²,

Thiéry³

2012

Ann. Appl. Probab.

103

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The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density. In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hilbert space. These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions. We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process. Our results imply that, in stationarity, the MALA algorithm applied to an N -dimensional approximation of the target will take O(N 1/3 ) steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require O(N ) steps when applied to the same class of problems. As a by-product of the diffusion limit, it also follows that the MALA algorithm is optimized at an average acceptance probability of 0.574. Previous results were proved only for targets which are products of one-dimensional distributions, or for variants of this situation, limiting their applicability. The correlation in our target means that the rescaled MALA algorithm converges weakly to an infinite dimensional Hilbert space valued diffusion, and the limit cannot be described through analysis of scalar diffusions. The limit theorem is proved by showing that a drift-martingale decomposition of the Markov chain, suitably scaled, closely resembles a weak Euler-Maruyama discretization of the putative limit. An invariance principle is proved for the martingale, and a continuous mapping argument is used to complete the proof.

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“…The same method of proof has also been applied to derive optimal scaling results for other types of Markov chain Monte Carlo algorithms, such as the Metropolis-adjusted Langevin algorithm; see Roberts & Rosenthal (1998, 2001, Breyer, Piccioni & Scarlatti (2002), Christensen, Roberts & Rosenthal (2003), Neal & Roberts (2006). In this paper, we consider Metropolis algorithms only; we do not give an exhaustive account of the literature about the Metropolis-adjusted Langevin algorithm.…”

Section: History Of Optimal Scalingmentioning

confidence: 99%

Optimal scaling of Metropolis algorithms: Heading toward general target distributions

Bédard¹,

Rosenthal

2008

Can J Statistics

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Abstract:The authors provide an overview of optimal scaling results for the Metropolis algorithm with Gaussian proposal distribution. They address in more depth the case of high-dimensional target distributions formed of independent, but not identically distributed components. They attempt to give an intuitive explanation as to why the well-known optimal acceptance rate of 0.234 is not always suitable. They show how to find the asymptotically optimal acceptance rate when needed, and they explain why it is sometimes necessary to turn to inhomogeneous proposal distributions. Their results are illustrated with a simple example. Echelonnage optimal de l'algorithme Metropolis : en route vers des lois cibles généralesRésumé : Les auteurs font un survol des résultats sur l'échelonnage optimal de l'algorithme Metropolis avec loi instrumentale gaussienne. Ils s'intéressent de plus près au cas des lois cibles multivariéesà composantes indépendantes mais non identiquement distribuées. Ils tentent d'expliquer en termes intuitifs pourquoi le taux d'acceptation optimal bien connu de 0.234 ne convient pas toujours. Ils montrent comment déterminer au besoin le taux d'acceptation asymptotiquement optimal et ils expliquent pourquoi il est parfois indispensable d'avoir recoursà des lois instrumentales non homogènes. Leurs résultats sont illustrés dans un cas simple.

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Optimal scaling of MaLa for nonlinear regression

Cited by 23 publications

References 15 publications

On the stability of sequential Monte Carlo methods in high dimensions

On the stability of sequential Monte Carlo methods in high dimensions

Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions

Optimal scaling of Metropolis algorithms: Heading toward general target distributions

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