turn-key and scalable synchronous distributed MCMC algorithm

Last week, I attended a Lagrange seminar where Vincent Plassier presented a ICML²¹ paper he had co-authored with Maxime Vono, Alain Durmus, and Eric Moulines. Aiming at distributed MCMC algorithms that operate on several machines, with a target distribution that breaks as a target

where θ is common to all terms. And each term in the product can (only) be computed locally. This setup is obviously the same as for the embarrassingly parallel approaches of Neiswanger et al. (2014) and Scott et al. (2016). And it follows an earlier proposal of Vono et al. (2020), which appears as a full Gibbs algorithm on the augmented parameters (θ,z), assuming each term is a conditional density in the latent z’s. Which requires constant communications between the b workers and the central “master” node when θ is concerned. The ICML²¹ paper overcomes this difficulty by defining an approximate target with a Normal component in z. Meaning that the (approximate) conditional distribution of θ given the latent z is Normal, i.e. considering the augmented joint

but despite the Gaussian aspect, this is not always practical:

“When d [is large], this Gibbs sampling scheme unfortunately leads to prohibitive computational costs and hence prevents its practical use for general Bayesian inference problems.”

The authors then move to simulating from several Langevin step, more specifically running one move of the Euler-Maruyama discretisation scheme of the overdamped Langevin stochastic differential equation. Communication with the central node is then reduced. The paper proposes a proof of convergence in this unusual (since overdamped) setup. As well as bounds on the bias due to the inclusion of the latent variables. They also manage to find the required scaling of the various parameters involved (Normal variance, discretisation scale, Langevin runs) to achieve convergence, which I find rather remarkable. The table at the top illustrates the comparison with earlier methods, whenever available.

far south

computational statistics and molecular simulation [18w5023]

This Thursday, our X fertilisation workshop at the interface between molecular dynamics and Monte Carlo statistical methods saw a wee bit of reduction in the audience as some participants had already left Oaxaca. Meaning they missed the talk of Christophe Andrieu on hypocoercivity which could have been another hand-on lecture, given the highly pedagogical contents of the talk. I had seen some parts of the talk in MCqMC 2018 in Rennes and at NUS, but still enjoyed the whole of it very much, and so did the audience given the induced discussion. For instance, previously, I had not seen the connection between the guided random walks of Gustafson and Diaconis, and continuous time processes like PDMP. Which Christophe also covered in his talk. (Also making me realise my colleague Jean Dolbeault in Dauphine was strongly involved in the theoretical analysis of PDMPs!) Then Samuel Power gave another perspective on PDMPs. With another augmentation, connected with time, what he calls trajectorial reversibility. This has the impact of diminishing the event rate, but creates some kind of reversibility which seems to go against the motivation for PDMPs. (Remember that all talks are available as videos on the BIRS webpage.) A remark in the talk worth reiterating is the importance of figuring out which kinds of approximations are acceptable in these approximations. Connecting somewhat with the next talk by Luc Rey-Bellet on a theory of robust approximations. In the sense of Poincaré, Gibbs, Bernstein, &tc. concentration inequalities and large deviations. With applications to rare events.The fourth and final “hand-on” session was run by Miranda Holmes-Certon on simulating under constraints. Motivated by research on colloids. For which the overdamp Langevin diffusion applies as an accurate model, surprisingly. Which makes a major change from the other talks [most of the workshop!] relying on this diffusion. (With an interesting intermede on molecular velcro made of DNA strands.) Connected with this example, exotic energy landscapes are better described by hard constraints. (Potentially interesting extension to the case when there are too many constraints to explore all of them?) Now, the definition of the measure projected on the manifold defined by the constraints is obviously an important step in simulating the distribution, which density is induced by the gradient of the constraints ∇q(x). The proposed algorithm is in the same spirit as the one presented by Tony the previous day, namely moving along the tangent space then on the normal space to get back to the manifold. A solution that causes issues when the gradient is (near) zero. A great hand-on session which induced massive feedback from the audience.

In the afternoon session, Gersende Fort gave a talk on a generalisation of the Wang-Landau algorithm, which modifies the true weights of the elements of a partition of the sampling space, to increase visits to low [probability] elements and jumps between modes. The idea is to rely on tempered versions of the original weights, learned by stochastic approximation. With an extra layer of adaptivity. Leading to an improvement with parameters that depends on the phase of the stochastic approximation. The second talk was by David Sanders on a recent paper in Chaos about importance sampling for rare events of (deterministic) billiard dynamics. With diffusive limits which tails are hard to evaluate, except by importance sampling. And the last talk of the day was by Anton Martinsson on simulated tempering for a molecular alignment problem. With weights of different temperatures proportional to the inverse of the corresponding normalising constants, which themselves can be learned by a form of bridge sampling  if I got it right.

On a very minor note, I heard at breakfast a pretty good story from a fellow participant having to give a talk at a conference that was moved to a very early time in the morning due to an official appearing at a later time and as a result “enjoying” a very small audience to the point that a cleaning lady appeared and started cleaning the board as she could not conceive the talks had already started! Reminding me of this picture at IHP.

computational statistics and molecular simulation [18w5023]

 I truly missed the gist of the first talk of the Wednesday morning of our X fertilisation workshop by Jianfeng Lu partly due to notations, although the topic very much correlated to my interests like path sampling, with an augmented version of HMC using an auxiliary indicator. And mentions made of BAOAB. Next, Marcello Pereyra spoke about Bayesian image analysis, with the difficulty of setting a prior on an image. In case of astronomical images there are motivations for an L¹ penalisation sparse prior. Sampling is an issue. Moreau-Yoshida proximal optimisation is used instead, in connection with our MCMC survey published in Stats & Computing two years ago. Transferability was a new concept for me, as introduced by Kerrie Mengersen (QUT), to extrapolate an estimated model to another system without using the posterior as a prior. With a great interlude about the crown of thorns starfish killer robot! Rather a prior determination based on historical data, in connection with recent (2018) Technometrics and Bayesian Analysis papers towards rejecting non-plausible priors. Without reading the papers (!), and before discussing the matter with Kerrie, here or in Marseille, I wonder at which level of precision this can be conducted. The use of summary statistics for prior calibration gave the approach an ABC flavour.

The hand-on session was Jonathan Mattingly’s discussion of gerrymandering reflecting on his experience at court! Hard to beat for an engaging talk reaching between communities. As it happens I discussed the original paper last year. Of course it was much more exciting to listen to Jonathan explaining his vision of the problem! Too bad I “had” to leave before the end for a [most enjoyable] rock climbing afternoon… To be continued at the dinner table! (Plus we got the complete explanation of the term gerrymandering, including this salamander rendering of the first identified as gerrymandered district!)

computational statistics and molecular simulation [18w5023]

This X fertilisation workshop Gabriel Stolz, Luke Bornn and myself organised towards reinforcing the interface between molecular dynamics and Monte Carlo statistical methods has now started! At the casa matematicà Oaxaca, the Mexican campus of BIRS, which is currently housed by a very nice hotel on the heights of Oaxaca. And after a fairly long flight for a large proportion of the participants. On the first day, Arthur Voter gave a fantastic “hand-on” review of molecular dynamics for material sciences, which was aimed at the statistician side of the audience and most helpful in my own understanding of the concepts and techniques at the source of HMC and PDMP algorithms. (Although I could not avoid a few mini dozes induced by jetlag.) Including the BAOAB version of HMC, which sounded to me like an improvement to investigate. The part on metastability, completed by a talk by Florian Maire, remained a wee bit mysterious [to me].

The shorter talks of the day all brought new perspectives and information to me (although they were definitely more oriented towards their “own” side of the audience than the hand-on lecture). For instance, Jesús María Sanz-Serna gave a wide ranging overview of numerical integrators and Tony Lelièvre presented a recent work on simulating measures supported by manifolds via an HMC technique constantly projecting over the manifold, with proper validation. (I had struggled with the paper this summer and this talk helped a lot.) There was a talk by Josh Fash on simulating implicit solvent models that mixed high-level programming and reversible jump MCMC, with an earlier talk by Yong Chen on variable dimension hidden Markov models that could have also alluded to reversible jump. Angela Bito talked about using ASIS (Ancillarity-sufficiency interweaving strategy) for improving the dynamics of an MCMC sampler associated with a spike & slab prior, the recentering-decentering cycle being always a sort of mystery to me [as to why it works better despite introducing multimodality in this case], and Gael Martin presented some new results on her on-going work with David Frazier about approximate Bayes with misspecified models, with the summary statistic being a score function that relates the work to the likelihood free approach of Bissiri et al.