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).

sampling, transport, and diffusions

This week, I am attending a very cool workshop at the Flatiron Institute (not in the Flatiron building!, but close enough) on Sampling, Transport, and Diffusions, organised by Bob Carpenter and Michael Albergo. It is quite exciting as I do not know most participants or their work! The Flatiron Institute is a private institute focussed on fundamental science funded by the Simons Foundation (in such working conditions universities cannot compete with!).

Eric Vanden-Eijden gave an introductory lecture on using optimal transport notion to improve sampling, with a PDE/ODE approach of continuously turning a base distribution into a target (formalised by the distribution at time one). This amounts to solving a velocity solution to an KL optimisation objective whose target value is zero. Velocity parameterised as a deep neural network density estimator. Using a score function in a reverse SDE inspired by Hyvärinnen (2005), with a surprising occurrence of Stein’s unbiased estimator, there for the same reasons of getting rid of an unknown element. In a lot of environments, simulating from the target is the goal and this can be achieved by MCMC sampling by normalising flows, learning the transform / pushforward map.

At the break, Yuling Yao made a very smart remark that testing between two models could also be seen as an optimal transport, trying to figure an optimal transform from one model to the next, rather than the bland mixture model we used in our mixtestin paper. At this point I have no idea about the practical difficulty of using / inferring the parameters of this continuum but one could start from normalising flows. Because of time continuity, one would need some driving principle.

Esteban Tabak gave another interest talk on simulating from a conditional distribution, which sounds like a no-problem when the conditional density is known but a challenge when only pairs are observed. The problem is seen as a transport problem to a barycentre obtained as a distribution independent from the conditioning z and then inverting. Constructing maps through flows. Very cool, even possibly providing an answer for causality questions.

Many of the transport talks involved normalizing flows. One by [Simons Fellow] Christopher Jazynski about adding to the Hamiltonian (in HMC) an artificial flow field  (Vaikuntanathan and Jarzynski, 2009) to make up for the Hamiltonian moving too fast for the simulation to keep track. Connected with Eric Vanden-Eijden’s talk in the end.

An interesting extension of delayed rejection for HMC by Chirag Modi, with a manageable correction à la Antonietta Mira. Johnatan Niles-Weed provided a nonparametric perspective on optimal transport following Hütter+Rigollet, 21 AoS. With forays into the Sinkhorn algorithm, mentioning Aude Genevay’s (Dauphine graduate) regularisation.

Michael Lindsey gave a great presentation on the estimation of the trace of a matrix by the Hutchinson estimator for sdp matrices using only matrix multiplication. Solution surprisingly relying on Gibbs sampling called thermal sampling.

And while it did not involve optimal transport, I gave a short (lightning) talk on our recent adaptive restore paper: although in retrospect a presentation of Wasserstein ABC could have been more suited to the audience.

day one at ISBA 22

Started the day with a much appreciated swimming practice in the [alas warm⁺⁺⁺] outdoor 50m pool on the Island with no one but me in the slooow lane. And had my first ride with the biXi system, surprised at having to queue behind other bikes at red lights! More significantly, it was a great feeling to reunite at last with so many friends I had not met for more than two years!!!

My friend Adrian Raftery gave the very first plenary lecture on his work on the Bayesian approach to long-term population projections, which was recently  a work censored by some US States, then counter-censored by the Supreme Court [too busy to kill Roe v. Wade!]. Great to see the use of Bayesian methods validated by the UN Population Division [with at least one branch of the UN

Stephen Lauritzen returning to de Finetti notion of a model as something not real or true at all, back to exchangeability. Making me wonder when exchangeability is more than a convenient assumption leading to the Hewitt-Savage theorem. And sufficiency. I mean, without falling into a Keynesian fallacy, each point of the sample has unique specificities that cannot be taken into account in an exchangeable model. Nice to hear some measure theory, though!!! Plus a comment on the median never being sufficient, recouping an older (and presumably not original) point of mine. Stephen’s (or Fisher’s?) argument being that the median cannot be recursively computed!

Antonietta Mira and I had our ABC session this afternoon with Cecilia Viscardi, Sirio Legramanti, and Massimiliano Tamborino (Warwick) as speakers. Cecilia linked ABC with normalising flows, in collaboration with Dennis Prangle (whose earlier paper on this connection was presented as the first One World ABC seminar). Thus using past simulations to approximate the posterior by a neural network, possibly with a significant increase in computing time when compared with more rudimentary SMC-ABC methods in larger dimensions. Sirio considered summary-free ABC based on discrepancies like Rademacher complexity. Which more or less contains MMD, Kullback-Leibler, Wasserstein and more, although it seems to be dependent on the parameterisation of the observations. An interesting opening at the end was that this approach could apply to non iid settings. Massi presented a paper coauthored with Umberto that had just been arXived. On sequential ABC with a dependence on the summary statistic (hence guided). Further bringing copulas into the game, although this forces another choice [for the marginals] in the method.

Tamara Broderick talked about a puzzling leverage effect of some observations in economic studies where a tiny portion of individuals may modify the significance or the sign of a coefficient, for which I cannot tell whether the data or the reliance on statistical significance are to blame. Robert Kohn presented mixture-of-Gaussian copulas [not to be confused with mixture of Gaussian-copulas!] and Nancy Reid concluded my first [and somewhat exhausting!] day at ISBA with a BFF talk on the different statistical paradigms take on confidence (for which the notion of calibration seems to remain frequentist).

Side comments: First, most people in the conference are wearing masks, which is great! Also, I find it hard to read slides from the screen, which I presume is an age issue (?!) Even more aside, I had Korean lunch in a place that refused to serve me a glass of water, which I find amazing.

accronyms [CDT lectures]

This week, I gave a short and introductory course in Warwick for the CDT (PhD) students on my perceived connections between reverse logistic regression à la Geyer and GANS, among other things. The first attempt was cancelled in 2020 due to the pandemic, the second one in 2021 was on-line and thus offered little possibilities for interactions. Preparing for this third attempt made me read more papers on some statistical analyses of GANs and WGANs, which was more satisfactory [for me] even though I could not get into the technical details…

GANs as density estimators

I recently read an arXival entitled Conditional Sampling With Monotone GAN by Kovakchi et al., who construct  a mapping T that transforms or pushes forward a reference measure þ(dθ) like a multivariate Normal distribution to a target conditional distribution ð(dθ|x).  Which makes the proposal a type of normalising flow, except it does not require a Jacobian derivation… The mapping T is monotonous and block triangular in order to be invertible. It is learned from data by minimising a functional divergence between Tþ(dθ) and ð(dθ|x), for instance GAN least square or GAN Wasserstein penalties and representing T as a neural network.  Where monotonicity is imposed by a Lagrangian. The authors “note that global minimizers of [their GAN criterion] can also be used for conditional density estimation” but I fail to understand the distinction in that once T is constructed, the estimated conditional density is automatically available. However my main source of puzzlement is at the worth of this construction, since it does not provide an exact generative process for the conditional distribution, while requiring many generations from the joint distribution. Rather than a comparison with MCMC, which is not applicable in untractable generative models, a comparison with less expensive ABC solutions would have been appropriate, I think. And the paper is missing any quantification on the quality or asymptotics of the density estimate provided by this involved approximation, as most of the recent literature on normalising flows and friends. (A point acknowledged by the authors in the supplementary material section.)

“In this regard, the MGANs approach introduced in the article belongs to the category of sampling techniques such as MCMC, whose goal is to generate independent samples from the law of y|x, as opposed to assuming some structural form of the probability measure directly.”

I am unsure I understand the above remark as MCMC methods are intrinsically linked with the exact probability distribution, exploiting either some conditional representations as in Gibbs or at the very least the ability to compute the joint density…