connection between tempering & entropic mirror descent

The next One World ABC webinar is this  Thursday,  the 2nd May, at 9am UK time, with Francesca Crucinio (King’s College London, formerly CREST and even more formerly Warwick) presenting

“A connection between Tempering and Entropic Mirror Descent”.

a joint work with Nicolas Chopin and Anna Korba (both from CREST) whose abstract follows:

This work explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC corresponds to entropic mirror descent applied to the reverse Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be seen as a descent scheme of the KL divergence with respect to the Fisher-Rao geometry, in contrast to Langevin dynamics that perform descent of the KL with respect to the Wasserstein-2 geometry. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and derive adaptive tempering rules that improve over other alternative benchmarks in the literature.

parallel tempering on optimised paths

Saifuddin Syed, Vittorio Romaniello, Trevor Campbell, and Alexandre Bouchard-Côté, whom I met and discussed with on my “last” trip to UBC, on December 2019, just arXived a paper on parallel tempering (PT), making the choice of tempering path an optimisation problem. They address the touchy issue of designing a sequence of tempered targets when the starting distribution π⁰, eg the prior, and the final distribution π¹, eg the posterior, are hugely different, eg almost singular.

“…theoretical analysis of reversible variants of PT has shown that adding too many intermediate chains can actually deteriorate performance (…) [while] on non reversible regime adding more chains is guaranteed to improve performances.”

The above applies to geometric combinations of π⁰ and π¹. Which “suffers from an arbitrarily suboptimal global communication barrier“, according to the authors (although the counterexample is not completely convincing since π⁰ and π¹ share the same variance). They propose a more non-linear form of tempering with constraints on the dependence of the powers on the temperature t∈(0,1).  Defining the global communication barrier as an average over temperatures of the rejection rate, the path characteristics (e.g., the coefficients of a spline function) can then be optimised in terms of this objective. And the temperature schedule is derived from the fact that the non-asymptotic round trip rate is maximized when the rejection rates are all equal. (As a side item, the technique exposed in the earlier tempering paper by Syed et al. was recently exploited for a night high resolution imaging of a black hole from the M87 galaxy.)

computational statistics and molecular simulation [18w5023]

On Day 2, Carsten Hartmann used a representation of the log cumulant as solution to a minimisation problem over a collection of importance functions (by the Vonsker-Varadhan principle), with links to X entropy and optimal control, a theme also considered by Alain Dunmus when considering the uncorrected discretised Langevin diffusion with a decreasing sequence of discretisation scale factors (Jordan, Kinderlehrer and Otto) in the spirit of convex regularisation à la Rockafellar. Also representing ULA as an inexact gradient descent algorithm. Murray Pollock (Warwick) presented a new technique called fusion to simulate from products of d densities, as in scalable MCMC (but not only). With an (early) starting and startling remark that when simulating one realisation from each density in the product and waiting for all of them to be equal means simulating from the product, in a strong link to the (A)BC fundamentals. This is of course impractical and Murray proposes to follow d Brownian bridges all ending up in the average of these simulations, constructing an acceptance probability that is computable and validating the output.

The second “hand-on” lecture was given by Gareth Roberts (Warwick) on the many aspects of scaling MCMC algorithms, which started with the famous 0.234 acceptance rate paper in 1996. While I was aware of some of these results (!), the overall picture was impressive, including a notion of complexity I had not seen before. And a last section on PDMPs where Gareth presented very recent on the different scales of convergence of Zigzag and bouncy particle samplers, mostly to the advantage of Zigzag.In the afternoon, Jeremy Heng presented a continuous time version of simulated tempering by adding a drift to the Langevin diffusion with time-varying energy, which must be solution to the Liouville pde . Which connects to a flow transport problem when solving the pde under additional conditions. Unclear to me was the creation of the infinite sequence. This talk was very much at the interface in the spirit of the workshop! (Maybe surprisingly complex when considering the endpoint goal of simulating from a given target.) Jonathan Weare’s talk was about quantum chemistry which translated into finding eigenvalues of an operator. Turning in to a change of basis in a inhumanly large space (10¹⁸⁰ dimensions!). Matt Moore presented the work on Raman spectroscopy he did while a postdoc at Warwick, with an SMC based classification of the peaks of a spectrum (to be used on Mars?) and Alessandra Iacobucci (Dauphine) showed us the unexpected thermal features exhibited by simulations of chains of rotors subjected to both thermal and mechanical forcings, which we never discussed in Dauphine beyond joking on her many batch jobs running on our cluster!

And I remembered today that there is currently and in parallel another BIRS workshop on statistical model selection [and a lot of overlap with our themes] taking place in Banff! With snow already there! Unfair or rather #unfair, as someone much too well-known would whine..! Not that I am in a position to complain about the great conditions here in Oaxaca (except for having to truly worry about stray dogs rather than conceptually about bears makes running more of a challenge, if not the altitude since both places are about the same).

nested sampling when prior and likelihood clash

A recent arXival by Chen, Hobson, Das, and Gelderblom makes the proposal of a new nested sampling implementation when prior and likelihood disagree, making simulations from the prior inefficient. The paper holds the position that a single given prior is used over and over all datasets that come along:

“…in applications where one wishes to perform analyses on many thousands (or even millions) of different datasets, since those (typically few) datasets for which the prior is unrepresentative can absorb a large fraction of the computational resources.” Chen et al., 2018

My reaction to this situation, provided (a) I want to implement nested sampling and (b) I realise there is a discrepancy, would be to resort to an importance sampling resolution, as we proposed in our Biometrika paper with Nicolas. Since one objection [from the authors] is that identifying outlier datasets is complicated (it should not be when the likelihood function can be computed) and time-consuming, sequential importance sampling could be implemented.

“The posterior repartitioning (PR) method takes advantage of the fact that nested sampling makes use of the likelihood L(θ) and prior π(θ) separately in its exploration of the parameter space, in contrast to Markov chain Monte Carlo (MCMC) sampling methods or genetic algorithms which typically deal solely in terms of the product.” Chen et al., 2018

The above salesman line does not ring a particularly convincing chime in that nested sampling is about as myopic as MCMC since based on the similar notion of a local proposal move, starting from the lowest likelihood argument (the minimum likelihood estimator!) in the nested sample.

“The advantage of this extension is that one can choose (π’,L’) so that simulating from π’ under the constraint L'(θ) > l is easier than simulating from π under the constraint L(θ) > l. For instance, one may choose an instrumental prior π’ such that Markov chain Monte Carlo steps adapted to the instrumental constrained prior are easier to implement than with respect to the actual constrained prior. In a similar vein, nested importance sampling facilitates contemplating several priors at once, as one may compute the evidence for each prior by producing the same nested sequence, based on the same pair (π’,L’), and by simply modifying the weight function.” Chopin & Robert, 2010

Since the authors propose to switch to a product (π’,L’) such that π’.L’=π.L, the solution appears like a special case of importance sampling, with the added drwaback that when π’ is not normalised, its normalised constant must be estimated as well. (With an extra nested sampling implementation?) Furthermore, the advocated solution is to use tempering, which is not so obvious as it seems in small dimensions. As the mass does not always diffuse to relevant parts of the space. A more “natural” tempering would be to use a subsample in the (sub)likelihood for nested sampling and keep the remainder of the sample for weighting the evaluation of the evidence.

accelerating MCMC

I have recently [well, not so recently!] been asked to write a review paper on ways of accelerating MCMC algorithms for the [review] journal WIREs Computational Statistics and would welcome all suggestions towards the goal of accelerating MCMC algorithms. Besides [and including more on]

• coupling strategies using different kernels and switching between them;
• tempering strategies using flatter or lower dimensional targets as intermediary steps, e.g., à la Neal;
• sequential Monte Carlo with particle systems targeting again flatter or lower dimensional targets and adapting proposals to this effect;
• Hamiltonian MCMC, again with connections to Radford (and more generally ways of avoiding rejections);
• adaptive MCMC, obviously;
• Rao-Blackwellisation, just as obviously (in the sense that increasing the precision in the resulting estimates means less simulations).