diffusions, sampling, and transport

The third and final day of the workshop was shortened for me as I had to catch an early flight back to Paris (and as I got overly conservative in my estimation for returning to JFK, catching a train with no delay at Penn Station and thus finding myself with two hours free before boarding, hence reviewing remaining Biometrika submission at the airport while waiting). As a result I missed the afternoon talks.

The morning was mostly about using scores for simulation (a topic of which I was mostly unaware), with Yang Song giving the introductory lecture on creating better [cf pix left] generative models via the score function, with a massive production of his on the topic (but too many image simulations of dogs, cats, and celebrities!). Estimating directly the score is feasible via Fisher divergence or score matching à la Hyvärinen (with a return of Stein’s unbiased estimator of the risk!). And relying on estimated scores to simulate / generate by Langevin dynamics or other MCMC methods that do not require density evaluations. Due to poor performances in low density / learning regions a fix is randomization / tempering but the resolution (as exposed) sounded clumsy. (And made me wonder at using some more advanced form of deconvolution since the randomization pattern is controlled.) The talk showed some impressive text to image simulations used by an animation studio!

And then my friend Arnaud Doucet continued on the same theme, motivating by estimating normalising constant through annealed importance sampling [Yuling’s meta-perspective comes back to mind in that the geometric mixture is not the only choice, but with which objective]. In AIS, as in a series of Arnaud’s works, like the 2006 SMC Read Paper with Pierre Del Moral and Ajay Jasra, the importance (!) of some auxiliary backward kernels goes beyond theoretical arguments, with the ideally sequence being provided by a Langevin diffusion. Hence involving a score, learned as in the previous talk. Arnaud reformulated this issue as creating a transportation map and its reverse, which is leading to their recent Schrödinger bridge generative model. Which [imho] both brings a unification perspective to his work and an efficient way to bridge prior to posterior in AIS. A most profitable morn for me!

Overall, this was an exhilarating workshop, full of discoveries for me and providing me with the opportunity to meet and exchange with mostly people I had not met before. Thanks to Bob Carpenter and Michael Albergo for organising and running the workshop!

transport, diffusions, and sampling

At the Sampling, Transport, and Diffusions workshop at the Flatiron Institute, on Day #2, Marilou Gabrié (École Polytechnique) gave the second introductory lecture on merging sampling and normalising flows targeting the target distribution, when driven by a divergence criterion like KL, that only requires the shape of the target density. I first wondered about ergodicity guarantees in simultaneous MCMC and map training due to the adaptation of the flow but the update of the map only depends on the current particle cloud in (8). From an MCMC perspective, it sounds somewhat paradoxical to see the independent sampler making such an unexpected come-back when considering that no insider information is available about the (complex) posterior to drive the [what-you-get-is-what-you-see] construction of the transport map. However, the proposed approach superposed local (random-walk like) and global (transport) proposals in Algorithm 1.

Qiang Liu followed on learning transport maps, with the  Interesting notion of causalizing a graph by removing intersections (which are impossible for an ODE, as discussed by Eric Vanden-Eijden’s talk yesterday) through  coupling. Which underlies his notion of rectified flows. Possibly connecting with the next lightning talk by Jonathan Weare on spurious modes created by a variational Monte Carlo sampler and the use of stochastic gradient, corrected by (case-dependent?) regularisation.

Then came a whole series of MCMC talks!

Sam Livingstone spoke on Barker’s proposal (an incoming Biometrika paper!) as part of a general class of transforms g of the MH ratio, using jump processes based on a nasty normalising constant related with g (tractable for the original Barker algorithm). I then realised I had missed his StatSci paper on how to speak to statistical physics researchers!

Charles Margossian spoke about using a massive number of short parallel runs (many-short-chain regime) from a recent paper written with Aki,  Andrew, and Lionel Riou-Durand (Warwick) among others. Which brings us back to the challenge of producing convergence diagnostics and precisely the Gelman-Rubin R statistic or its recent nR avatar (with its linear limitations and dependence on parameterisation, as opposed to fuller distributional criteria). The core of the approach is in using blocks of GPUs to improve and speed-up the estimation of the between-chain variance. (D for R².) I still wonder at a waste of simulations / computing power resulting from stopping the runs almost immediately after warm-up is over, since reaching the stationary regime or an approximation thereof should be exploited more efficiently. (Starting from a minimal discrepancy sample would also improve efficiency.)

Lu Zhang also talked on the issue of cutting down warmup, presenting a paper co-authored with Bob, Andrew, and Aki, recommending Laplace / variational approximations for reaching faster high-posterior-density regions, using an algorithm called Pathfinder that relies on ELBO checks to counter poor performances of Laplace approximations. In the spirit of the workshop, it could be profitable to further transform / push-forward the outcome by a transport map.

Yuling Yao (of stacking and Pareto smoothing fame!) gave an original and challenging (in a positive sense) talk on the many ways of bridging densities [linked with the remark he shared with me the day before] and their statistical significance. Questioning our usual reliance on arithmetic or geometric mixtures. Ignoring computational issues, selecting a bridging pattern sounds not different from choosing a parameterised family of embedding distributions. This new typology of models can then be endowed with properties that are more or less appealing. (Occurences of the Hyvärinen score and our mixtestin perspective in the talk!)

Miranda Holmes-Cerfon talked about MCMC on stratification (illustrated by this beautiful picture of nanoparticle random walks). Which means sampling under varying constraints and dimensions with associated densities under the respective Hausdorff measures. This sounds like a perfect setting for reversible jump and in a sense it is, as mentioned in the talks. Except that the moves between manifolds are driven by the proximity to said manifold, helping with a higher acceptance rate, and making the proposals easier to construct since projections (or the reverses) have a physical meaning. (But I could not tell from the talk why the approach was seemingly escaping the symmetry constraint set by Peter Green’s RJMCMC on the reciprocal moves between two given manifolds).

diffusion means in geometric statistics [CRiSM Seminar]

MCMskv #4 [house with a vision]

Last day at MCMskv! Not yet exhausted by this exciting conference, but this was the toughest day with one more session and a tutorial by Art Own on quasi Monte-Carlo. (Not even mentioning the night activities that I skipped. Or the ski break that I did not even consider.) Krys Latunszynski started with a plenary on exact methods for discretised diffusions, with a foray in Bernoulli factory problems. Then a neat session on adaptive MCMC methods that contained a talk by Chris Sherlock on delayed acceptance, where the approximation to the target was built by knn trees. (The adaptation was through the construction of the tree by including additional evaluations of the target density. Another paper sitting in my to-read list for too a long while: the exploitation of the observed values of π towards improving an MCMC sampler has always be “obvious” to me even though I could not see any practical way of doing so. )

It was wonderful that Art Owen accepted to deliver a tutorial at MCMskv on quasi-random Monte Carlo. Great tutorial, with a neat coverage of the issues most related to Monte Carlo integration. Since quasi-random sequences have trouble with accept/reject methods, a not-even-half-baked idea that came to me during Art’s tutorial was that the increased computing power granted by qMC could lead to a generic integration of the Metropolis-Hastings step in a Rao-Blackwellised manner. Art mentioned he was hoping that in a near future one could switch between pseudo- and quasi-random in an almost automated manner when running standard platforms like R. This would indeed be great, especially since quasi-random sequences seem to be available at the same cost as their pseudo-random counterpart. During the following qMC session, Art discussed the construction of optimal sequences on sets other than hypercubes (with the surprising feature that projecting optimal sequences from the hypercube does not work). Mathieu Gerber presented the quasi-random simulated annealing algorithm he developed with Luke Bornn that I briefly discussed a while ago. Or thought I did as I cannot trace a post on that paper! While the fact that annealing also works with quasi-random sequences is not astounding, the gain over random sequences shown on two examples is clear. The session also had a talk by Lester Mckey who relies Stein’s discrepancy to measure the value of an approximation to the true target. This was quite novel, with a surprising connection to Chris Oates’ talk and the use of score-based control variates, if used in a dual approach.

Another great session was the noisy MCMC one organised by Paul Jenkins (Warwick), with again a coherent presentation of views on the quality or lack thereof of noisy (or inexact) versions, with an update from Richard Everitt on inexact MCMC, Felipe Medina Aguayo (Warwick) on sufficient conditions for noisy versions to converge (and counterexamples), Jere Koskela (Warwick) on a pseudo-likelihood approach to the highly complex Kingman’s coalescent model in population genetics (of ABC fame!), and Rémi Bardenet on the tall data approximations techniques discussed in a recent post. Having seen or read most of those results previously did not diminish the appeal of the session.

MCMC at ICMS (2)

The second day of our workshop on computational statistics at the ICMS started with a terrific talk by Xiao-Li Meng. Although this talk related with his Inception talk in Paris last summer, and of the JCGS discussion paper, he brought new geometric aspects to the phenomenon (managing a zero correlation and hence i.i.d.-ness in the simulation of a Gaussian random effect posterior distribution). While I was reflecting about the difficulty to extend the perspective beyond normal models, he introduced a probit example where exact null correlation cannot be found but an adaptive scheme allows to explore the range of correlation coefficients. This made me somehow think of a possible version in this approach in a tempering perspective, where different data augmentation schemes would be merged into an “optimal” geometric mixture, rather than via interweaving.

As an aside, Xiao-Li mentioned the idea of Bayesian sufficiency and Bayesian ancilarity in the construction of his data augmentation schemes. He then concluded that sufficiency is identical in classical and Bayesian approaches, while ancilarity could be defined in several ways. I have already posted on that, but it seems to me that sufficiency is a weaker notion in the Bayesian perspective in the sense that all that matters is that the posterior is the same given the observation y and given the observed statistics, rather than uniformly over all possible values of the random variable Y as in the classical sense. As for ancilarity, it is also natural to consider that an ancillary statistics does not bring information on the parameter, i.e. that the prior and the posterior distributions are the same given the observed ancillary statistics. Going further to define ancilarity as posterior independence between “true” parameters and auxiliary variables, as Xiao-Li suggested, does not seem very sound as it leads to the paradoxes Basu liked so much!

Today, the overlap with the previous meetings in Bristol and in Banff was again limited: Arnaud Doucet rewrote his talk towards less technicity, which means I got the idea much more clearly than last week. The idea of having a sequence of pseudo-parameters with the same pseudo-prior seems to open a wide range of possible adaptive schemes. Faming Liang also gave a talk fairly similar to the one he presented in Banff. And David van Dyk as well, which led me to think anew about collapsed Gibbs samplers in connection with ABC and a project I just started here in Edinburgh.

Otherwise, the intense schedule of the day saw us through eleven talks. Daniele Impartato called for distributions (in the physics or Laurent Schwarz’ meaning of the term!) to decrease the variance of Monte Carlo estimations, an approach I hope to look further as Schwarz’ book is the first math book I ever bought!, an investment I tried to capitalize once in writing a paper mixing James-Stein estimation and distributions for generalised integration by part, paper that was repeatedly rejected until I gave up! Jim Griffin showed us improvements brought in the exploration of large number of potential covariates in linear and generalised linear models. Natesh Pillai tried to drag us through several of his papers on covariance matrix estimation, although I fear he lost me along the way! Let me perversely blame the schedule (rather than an early rise to run around Arthur’s Seat!) for falling asleep during Alex Beskos’ talk on Hamiltonian MCMC for diffusions, even though I was looking forward this talk. (Apologies to Alex!) Then Simon Byrne gave us a quick tour of differential geometry in connection with orthogonalization for Hamiltonian MCMC. Which brought me back very briefly to this early time I was still considering starting a PhD in differential geometry and then even more briefly played with the idea of mixing differential geometry and statistics à la Shun’ichi  Amari…. Ian Murray and  Simo Sarkka completed the day with a cartoonesque talk on latent Gaussians that connected well with Xiao-Li’s and a talk on Gaussian approximations to diffusions with unknown parameters, which kept within the main theme of the conference, namely inference on partly observed diffusions.

As written above, this was too intense a day, with hardly any free time to discuss about the talks or the ongoing projects, which makes me prefer the pace adopted in Bristol or in Banff. Having to meet a local student on leave from Dauphine for a year here did not help of course!)