Archive for Langevin MCMC algorithm

scalable Metropolis-Hastings

Posted in Books, Statistics, Travel with tags , , , , , , , , , on February 12, 2019 by xi'an

Among the flury of arXived papers of last week (414!), including a fair chunk of papers submitted to ICML 2019, I spotted one entry by Cornish et al. on scalable Metropolis-Hastings, which Arnaud Doucet had mentioned to me yesterday when in Oxford. The paper builds on the delayed acceptance paper we wrote with Marco Banterlé, Clara Grazian and Anthony Lee, itself relying on a factorisation decomposition of the likelihood, combined with control variate accelerating techniques. The factorisation of both the target and the proposal allows for a (less efficient) Metropolis-Hastings acceptance ratio that is the product

\prod_{i=1}^m \alpha_i(\theta,\theta')

of individual Metropolis-Hastings acceptance ratios, but which allows for quicker rejection if one of the probabilities in the product is small, because the corresponding Bernoulli draw is zero with high probability. One advance made in Michel et al. (2017) [which I doubly missed] is that subsampling is achievable by thinning (as in PDMPs, where these authors have been quite active) through an algorithm of Shantikumar (1985) [described in Devroye’s bible]. Provided each Metropolis-Hastings probability can be lower bounded:

\alpha_i(\theta,\theta') \ge \exp\{-\psi_i \phi(\theta,\theta')\}

by a term where the transition φ does not depend on the index i in the product. The computing cost of the thinning process thus depends on the efficiency of the subsampling, namely whether or not the (Poisson) number of terms is much smaller than m, number of terms in the product. A neat trick in the current paper that extends the the Fukui-Todo procedure is to switch to the original Metropolis-Hastings when the overall lower bound is too small, recovering the geometric ergodicity of this original if it holds (Theorem 2.1). Another neat remark is that when using the naïve factorisation as the product of the n individual likelihoods, the resulting algorithm is sort of doomed as n grows, even with an optimal scaling of the proposals. To achieve scalability, the authors introduce a Taylor (i.e., Gaussian) approximation to each local target in the product and start the acceptance decomposition by using the resulting overall Gaussian approximation. Meaning that the remaining product is now made of ratios of targets over their local Taylor approximations, hence most likely close to one. And potentially lower-bounded by the remainder term in the Taylor expansion. Leading to the conclusion that, when everything goes well, meaning that the Taylor expansions can be conducted and the bounds derived for the appropriate expansion, the order of the Poisson scale is O(1/√n)..! The proposal for the Metropolis-Hastings move is actually tuned to the Gaussian approximation, appearing as a variant of the Langevin move or more exactly a discretization of an Hamiltonian move. Obviously, I cannot judge of the complexity in implementing this new scheme from just reading the paper, but this development on the split target is definitely an exciting prospect for handling huge datasets and their friends!

Markov chain importance sampling

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , on May 31, 2018 by xi'an

Ingmar Schuster (formerly a postdoc at Dauphine and now in Freie Universität Berlin) and Ilja Klebanov (from Berlin) have recently arXived a paper on recycling proposed values in [a rather large class of] Metropolis-Hastings and unadjusted Langevin algorithms. This means using the proposed variates of one of these algorithms as in an importance sampler, with an importance weight going from the target over the (fully conditional) proposal to the target over the marginal stationary target. In the Metropolis-Hastings case, since the later is not available in most setups, the authors suggest using a Rao-Blackwellised nonparametric estimate based on the entire MCMC chain. Or a subset.

“Our estimator refutes the folk theorem that it is hard to estimate [the normalising constant] with mainstream Monte Carlo methods such as Metropolis-Hastings.”

The paper thus brings an interesting focus on the proposed values, rather than on the original Markov chain,  which naturally brings back to mind the derivation of the joint distribution of these proposed values we made in our (1996) Rao-Blackwellisation paper with George Casella. Where we considered a parametric and non-asymptotic version of this distribution, which brings a guaranteed improvement to MCMC (Metropolis-Hastings) estimates of integrals. In subsequent papers with George, we tried to quantify this improvement and to compare different importance samplers based on some importance sampling corrections, but as far as I remember, we only got partial results along this way, and did not cover the special case of the normalising constant Þ… Normalising constants did not seem such a pressing issue at that time, I figure. (A Monte Carlo 101 question: how can we be certain the importance sampler offers a finite variance?)

Ingmar’s views about this:

I think this is interesting future work. My intuition is that for Metropolis-Hastings importance sampling with random walk proposals, the variance is guaranteed to be finite because the importance distribution ρ_θ is a convolution of your target ρ with the random walk kernel q. This guarantees that the tails of ρ_θ are no lighter than those of ρ. What other forms of q mean for the tails of ρ_θ I have less intuition about.

When considering the Langevin alternative with transition (4), I was first confused and thought it was incorrect for moving from one value of Y (proposal) to the next. But that’s what unadjusted means in “unadjusted Langevin”! As pointed out in the early Langevin literature, e.g., by Gareth Roberts and Richard Tweedie, using a discretised Langevin diffusion in an MCMC framework means there is a risk of non-stationarity & non-ergodicity. Obviously, the corrected (MALA) version is more delicate to approximate (?) but at the very least it ensures the Markov chain does not diverge. Even when the unadjusted Langevin has a stationary regime, its joint distribution is likely quite far from the joint distribution of a proper discretisation. Now this also made me think about a parameterised version in the 1996 paper spirit, but there is nothing specific about MALA that would prevent the implementation of the general principle. As for the unadjusted version, the joint distribution is directly available.  (But not necessarily the marginals.)

Here is an answer from Ingmar about that point

Personally, I think the most interesting part is the practical performance gain in terms of estimation accuracy for fixed CPU time, combined with the convergence guarantee from the CLT. ULA was particularly important to us because of the papers of Arnak Dalalyan, Alain Durmus & Eric Moulines and recently from Mike Jordan’s group, which all look at an unadjusted Langevin diffusion (and unimodal target distributions). But MALA admits a Metropolis-Hastings importance sampling estimator, just as Random Walk Metropolis does – we didn’t include MALA in the experiments to not get people confused with MALA and ULA. But there is no delicacy involved whatsoever in approximating the marginal MALA proposal distribution. The beauty of our approach is that it works for almost all Metropolis-Hastings algorithms where you can evaluate the proposal density q, there is no constraint to use random walks at all (we will emphasize this more in the paper).

JSM 2015 [day #4]

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , on August 13, 2015 by xi'an

My first session today was Markov Chain Monte Carlo for Contemporary Statistical Applications with a heap of interesting directions in MCMC research! Now, without any possible bias (!), I would definitely nominate Murray Pollock (incidentally from Warwick) as the winner for best slides, funniest presentation, and most enjoyable accent! More seriously, the scalable Langevin algorithm he developed with Paul Fearnhead, Adam Johansen, and Gareth Roberts, is quite impressive in avoiding computing costly likelihoods. With of course caveats on which targets it applies to. Murali Haran showed a new proposal to handle high dimension random effect models by a projection trick that reduces the dimension. Natesh Pillai introduced us (or at least me!) to a spectral clustering that allowed for an automated partition of the target space, itself the starting point to his parallel MCMC algorithm. Quite exciting, even though I do not perceive partitions as an ideal solution to this problem. The final talk in the session was Galin Jones’ presentation of consistency results and conditions for multivariate quantities which is a surprisingly unexplored domain. MCMC is still alive and running!

The second MCMC session of the morning, Monte Carlo Methods Facing New Challenges in Statistics and Science, was equally diverse, with Lynn Kuo’s talk on the HAWK approach, where we discovered that harmonic mean estimators are still in use, e.g., in MrBayes software employed in phylogenetic inference. The proposal to replace this awful estimator that should never be seen again (!) was rather closely related to an earlier solution of us for marginal likelihood approximation, based there on a partition of the whole space rather than an HPD region in our case… Then, Michael Betancourt brilliantly acted as a proxy for Andrew to present the STAN language, with a flashy trailer he most recently designed. Featuring Andrew as the sole actor. And with great arguments for using it, including the potential to run expectation propagation (as a way of life). In fine, Faming Liang proposed a bootstrap subsampling version of the Metropolis-Hastings algorithm, where the likelihood acknowledging the resulting bias in the limiting distribution.

My first afternoon session was another entry on Statistical Phylogenetics, somewhat continued from yesterday’s session. Making me realised I had not seen a single talk on ABC for the entire meeting! The issues discussed in the session were linked with aligning sequences and comparing  many trees. Again in settings where likelihoods can be computed more or less explicitly. Without any expertise in the matter, I wondered at a construction that would turn all trees, like  into realizations of a continuous model. For instance by growing one branch at a time while removing the MRCA root… And maybe using a particle like method to grow trees. As an aside, Vladimir Minin told me yesterday night about genetic mutations that could switch on and off phenotypes repeatedly across generations… For instance  the ability to glow in the dark for species of deep sea fish.

When stating that I did not see a single talk about ABC, I omitted Steve Fienberg’s Fisher Lecture R.A. Fisher and the Statistical ABCs, keeping the morceau de choix for the end! Even though of course Steve did not mention the algorithm! A was for asymptotics, or ancilarity, B for Bayesian (or biducial??), C for causation (or cuffiency???)… Among other germs, I appreciated that Steve mentioned my great-grand father Darmois in connection with exponential families! And the connection with Jon Wellner’s LeCam Lecture from a few days ago. And reminding us that Savage was a Fisher lecturer himself. And that Fisher introduced fiducial distributions quite early. And for defending the Bayesian perspective. Steve also set some challenges like asymptotics for networks, Bayesian model assessment (I liked the notion of stepping out of the model), and randomization when experimenting with networks. And for big data issues. And for personalized medicine, building on his cancer treatment. No trace of the ABC algorithm, obviously, but a wonderful Fisher’s lecture, also most obviously!! Bravo, Steve, keep thriving!!!

gradient importance sampling

Posted in Books, pictures, Statistics, University life with tags , , , , , , on July 30, 2015 by xi'an

from my office, La Défense & Bois de Boulogne, Paris, May 15, 2012Ingmar Schuster, who visited Paris-Dauphine last Spring (and is soon to return here as a postdoc funded by Fondation des Sciences Mathématiques de Paris) has arXived last week a paper on gradient importance sampling. In this paper, he builds a sequential importance sampling (or population Monte Carlo) algorithm that exploits the additional information contained in the gradient of the target. The proposal or importance function being essentially the MALA move as its proposal, mixed across the elements of the previous population. When compared with our original PMC mixture of random walk proposals found in e.g. this paper, each term in the mixture thus involves an extra gradient, with a scale factor that decreases to zero as 1/t√t. Ingmar compares his proposal with an adaptive Metropolis, an adaptive MALTa and an HM algorithms, for two mixture distributions and the banana target of Haario et al. (1999) we also used in our paper. As well as a logistic regression. In each case, he finds both a smaller squared error and a smaller bias for the same computing time (evaluated as the number of likelihood evaluations). While we discussed this scheme when he visited, I remain intrigued as to why it works so well when compared with the other solutions. One possible explanation is that the use of the gradient drift is more efficient on a population of particles than on a single Markov chain, provided the population covers all modes of importance on the target surface: the “fatal” attraction of the local model is then much less of an issue…

Cancún, ISBA 2014 [day #3]

Posted in pictures, Statistics, Travel, University life with tags , , , , , on July 23, 2014 by xi'an

Cancun13…already Thursday, our [early] departure day!, with an nth (!) non-parametric session that saw [the newly elected ISBA Fellow!] Judith Rousseau present an ongoing work with Chris Holmes on the convergence or non-convergence conditions for a Bayes factor of a non-parametric hypothesis against another non-parametric. I wondered at the applicability of this test as the selection criterion in ABC settings, even though having an iid sample to start with is a rather strong requirement.

Switching between a scalable computation session with Alex Beskos, who talked about adaptive Langevin algorithms for differential equations, and a non-local prior session, with David Rossell presenting a smoother way to handle point masses in order to accommodate frequentist coverage. Something we definitely need to discuss the next time I am in Warwick! Although this made me alas miss both the first talk of the non-local session by Shane Jensen  the final talk of the scalable session by Doug Vandewrken where I happened to be quoted (!) for my warning about discretising Markov chains into non-Markov processes. In the 1998 JASA paper with Chantal Guihenneuc.

After a farewell meal of ceviche with friends in the sweltering humidity of a local restaurant, I attended [the newly elected ISBA Fellow!] Maria Vanucci’s talk on her deeply involved modelling of fMRI. The last talk before the airport shuttle was François Caron’s description of a joint work with Emily Fox on a sparser modelling of networks, along with an auxiliary variable approach that allowed for parallelisation of a Gibbs sampler. François mentioned an earlier alternative found in machine learning where all components of a vector are updated simultaneously conditional on the previous avatar of the other components, e.g. simulating (x’,y’) from π(x’|y) π(y’|x) which does not produce a convergent Markov chain. At least not convergent to the right stationary. However, running a quick [in-flight] check on a 2-d normal target did not show any divergent feature, when compared with the regular Gibbs sampler. I thus wonder at what can be said about the resulting target or which conditions are need for divergence. A few scribbles later, I realised that the 2-d case was the exception, namely that the stationary distribution of the chain is the product of the marginal. However, running a 3-d example with an auto-exponential distribution in the taxi back home, I still could not spot a difference in the outcome.

a day for comments

Posted in Mountains, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , on April 21, 2014 by xi'an

As I was flying over Skye (with [maybe] a first if hazy perspective on the Cuillin ridge!) to Iceland, three long sets of replies to some of my posts appeared on the ‘Og:

Thanks to them for taking the time to answer my musings…


shrinkage-thresholding MALA for Bayesian variable selection

Posted in Statistics, University life with tags , , , , , on March 10, 2014 by xi'an

IMG_2515Amandine Shreck along with her co-authors Gersende Fort, Sylvain LeCorff, and Eric Moulines, all from Telecom Paristech, has undertaken to revisit the problem of large p small n variable selection. The approach they advocate mixes Langevin algorithms with trans-model moves with shrinkage thresholding. The corresponding Markov sampler is shown to be geometrically ergodic, which may be a première in that area. The paper was arXived in December but I only read it on my flight to Calgary, not overly distracted by the frozen plains of Manitoba and Saskatchewan. Nor by my neighbour watching Hunger Games II.)

A shrinkage-thresholding operator is defined as acting on the regressor matrix towards producing sparse versions of this matrix. (I actually had trouble picturing the model until Section 2.2 where the authors define the multivariate regression model, making the regressors a matrix indeed. With a rather unrealistic iid Gaussian noise. And with an unknown number of relevant rows, hence a varying dimension model. Note that this is a strange regression in that the regression coefficients are known and constant across all models.) Because the Langevin algorithm requires a gradient to operate, the log target is divided between a differentiable and a non-differentiable parts, the later accommodating the Dirac masses in the dominating measure. The new MALA moves involve applying the above shrinkage-thresholding operator to a regular Langevin proposal, hence moving to sub-spaces and sparser representations.

The thresholding functions are based on positive part operators, which means that the Markov chain does not visit some neighbourhoods of zero in the embedding and in the sparser spaces. In other words, the proposal operates between models of varying dimensions without further ado because the point null hypotheses are replaced with those neighbourhoods. Hence it is not exactly simulating from the “original” posterior, which may be a minor caveat or not. Not if defining the neighbourhoods is driven by an informed or at least spelled-out choice of a neighbourhood of zero where the coefficients are essentially identified with zero. The difficulty is then in defining how close is close enough. Especially since the thresholding functions seem to all depend on a single number which does not depend on the regressor matrix. It would be interesting to see if the g-prior version could be developed as well… Actually, I would have also included a dose of g-prior in the Langevin move, rather than using an homogeneous normal noise.

The paper contains a large experimental part where the performances of the method are evaluated on various simulated datasets. It includes a comparison with reversible jump MCMC, which slightly puzzles me: (a) I cannot see from the paper whether or not the RJMCMC is applied to the modified (thresholded) posterior, as a regular RJMCMC would not aim at the same target, but the appendix does not indicate a change of target; (b) the mean error criterion for which STMALA does better than RJMCMC is not defined, but the decrease of this criterion along iterations seems to indicate that convergence has not yet occured, since it does not completely level up after 3 10⁵ iterations.

I must have mentioned it in another earlier post, but I find somewhat ironical to see those thresholding functions making a comeback after seeing the James-Stein and smooth shrinkage estimators taking over the then so-called pre-test versions in the 1970’s (Judge and Bock, 1978) and 1980’s. There are obvious reasons for this return, moving away from quadratic loss being one.