## Archive for CREST

## Langevin on a wrong bend

Posted in Books, Statistics with tags CREST, Hastings-Metropolis sampler, Langevin diffusion, MALA, pseudo-marginal MCMC, scalable MCMC, stochastic gradient descent, Wasserstein distance on October 19, 2017 by xi'an**A**rnak Dalayan and Avetik Karagulyan (CREST) arXived a paper the other week on a focussed study of the Langevin algorithm [not MALA] when the gradient of the target is incorrect. With the following improvements *[quoting non-verbatim from the paper]*:

- a varying-step Langevin that reduces the number of iterations for a given Wasserstein precision, compared with recent results by e.g. Alan Durmus and Éric Moulines;
- an extension of convergence results for error-prone evaluations of the gradient of the target (i.e., the gradient is replaced with a noisy version, under some moment assumptions that do not include unbiasedness);
- a new second-order sampling algorithm termed LMCO’, with improved convergence properties.

What is particularly interesting to me in this setting is the use in all these papers of a discretised Langevin diffusion (a.k.a., random walk with a drift induced by the gradient of the log-target) without the original Metropolis correction. The results rely on an assumption of [strong?] log-concavity of the target, with “user-friendly” bounds on the Wasserstein distance depending on the constants appearing in this log-concavity constraint. And so does the adaptive step. (In the case of the noisy version, the bias and variance of the noise also matter. As pointed out by the authors, there is still applicability to scaling MCMC for large samples. Beyond pseudo-marginal situations.)

“…this, at first sight very disappointing behavior of the LMC algorithm is, in fact, continuously connected to the exponential convergence of the gradient descent.”

The paper concludes with an interesting mise en parallèle of Langevin algorithms and of gradient descent algorithms, since the convergence rates are the same.

## probably ABC [and provably robust]

Posted in Books, pictures, Statistics, Travel with tags ABC, ABC-SMC, adaptive Monte Carlo algorithm, Bayesian asymptotics, CREST, Gaussian processes, likelihood-free methods, misspecified model, oracle inequalities on August 8, 2017 by xi'an**T**wo weeks ago, James Ridgway (formerly CREST) arXived a paper on misspecification and ABC, a topic on which David Frazier, Judith Rousseau and I have been working for a while now [and soon to be arXived as well]. Paper that I re-read on a flight to Amsterdam [hence the above picture], written as a continuation of our earlier paper with David, Gael, and Judith. One specificity of the paper is to use an exponential distribution on the distance between the observed and simulated sample within the ABC distribution. Which reminds me of the resolution by Bissiri, Holmes, and Walker (2016) of the intractability of the likelihood function. James’ paper contains oracle inequalities between the ABC approximation and the genuine distribution of the summary statistics, like a bound on the distance between the expectations of the summary statistics under both models. Which writes down as a sum of a model bias, of two divergences between empirical and theoretical averages, on smoothness penalties, and on a prior impact term. And a similar bound on the distance between the expected distance to the oracle estimator of θ under the ABC distribution [and a Lipschitz type assumption also found in our paper]. Which first sounded weird [to me] as I would have expected the true posterior, until it dawned on me that the ABC distribution is the one used for the estimation [a passing strike of over-Bayesianism!]. While the oracle bound could have been used directly to discuss the rate of convergence of the exponential rate λ to zero [with the sample size n], James goes into the interesting alternative direction of setting a prior on λ, an idea that dates back to Olivier Catoni and Peter Grünwald. Or rather a pseudo-posterior on λ, a common occurrence in the PAC-Bayesian literature. In one of his results, James obtains a dependence of λ on the dimension m of the summary [as well as the root dependence on the sample size n], which seems to contradict our earlier independence result, until one realises this scale parameter is associated with a distance variable, itself scaled in m.

The paper also contains a non-parametric part, where the parameter θ is the unknown distribution of the data and the summary the data itself. Which is quite surprising as I did not deem it possible to handle non-parametrics with ABC. Especially in a misspecified setting (although I have trouble perceiving what this really means).

“We can use most of the Monte Carlo toolbox available in this context.”

The theoretical parts are a bit heavy on notations and hard to read [as a vacation morning read at least!]. They are followed by a Monte Carlo implementation using SMC-ABC. And pseudo-marginals [at least formally as I do not see how the specific features of pseudo-marginals are more that an augmented representation here]. And adaptive multiple pseudo-samples that reminded me of the Biometrika paper of Anthony Lee and Krys Latuszynski (Warwick). Therefore using indeed most of the toolbox!

## end of a long era [1982-2017]

Posted in Books, pictures, Running, University life with tags École Polytechnique, badge, boxes, CREST, ENSAE, INSEE, Insee Paris Club, Malakoff, moving, office, Paris, Paris-Saclay campus on May 23, 2017 by xi'an**T**his afternoon I went to CREST to empty my office there from books and a few papers (like the original manuscript version of Monte Carlo Statistical Methods). This is because the research centre, along with the ENSAE graduate school (my Alma mater), is moving to a new building on the Saclay plateau, next to École Polytechnique. As part of this ambitious migration of engineering schools from downtown Paris to a brand new campus there. Without getting sentimental about this move, it means leaving the INSEE building in Malakoff, on the outskirts of downtown Paris, which has been an enjoyable part of my student and then academic life from 1982 till now. And also leaving the INSEE Paris Club runners! (I am quite uncertain about being as active at the new location, if only because going there by bike is a bit more of a challenge. To be addressed anyway!) And I left behind my accumulation of conference badges (although I should try to recycle them for the incoming BNP 11 in Paris!).

## Alan Gelfand in Paris

Posted in pictures, Statistics, Travel, University life with tags Alan Gelfand, BiPS, CREST, Duke University, ENSAE, Gaussian processes, Paris, seminar on May 11, 2017 by xi'anAlan Gelfand (Duke University) will be in Paris on the week of May 15 and give several seminars, including one at AgroParisTech on May 16:

and on at CREST (BiPS) on May 18, 2pm:

Scalable Gaussian processes for analyzing space and space-time datasets

## truncated normal algorithms

Posted in Books, pictures, R, Statistics, University life with tags CREST, Nicolas Chopin, Statisfaction, truncated normal, Utah on January 4, 2017 by xi'an**N**icolas Chopin (CREST) just posted an entry on Statisfaction about the comparison of truncated Normal algorithms run by Alan Rogers, from the University of Utah. Nicolas wrote a paper in Statistics and Computing about a simulation method, which proposes a Ziggurat type of algorithm for this purpose, and which I do not remember reading, thanks to my diminishing memory buffer! As shown in the picture below, when truncating to the half-line (a,∞), this method improves upon my accept-reject approach except in the far tails.

On the top graph, made by Alan Rogers, my uniform proposal (r) seems to be doing better for a Normal truncated to (a,b) when b<0, or when a gets large and close to b. Nicolas’ ziggurat (c) works better than the Gaussian accept-reject method (c) on the positive part. (I wonder what the exponential proposal (e) stands for, in terms of scale parameter.)

## zig, zag, and subsampling

Posted in Books, Statistics, University life with tags BiPS, CREST, ENSAE, Malakoff, MCMC, Monte Carlo Statistical Methods, Paris, Université Paris Dauphine, University of Warwick, Zig-Zag on December 29, 2016 by xi'an**T**oday, I alas missed a seminar at BiPS on the Zig-Zag (sub-)sampler of Joris Bierkens, Paul Fearnhead and Gareth Roberts, presented here in Paris by James Ridgway. Fortunately for me, I had some discussions with Murray Pollock in Warwick and then again with Changye Wu in Dauphine that shed some light on this complex but highly innovative approach to simulating in Big Data settings thanks to a correct subsampling mechanism.

The zig-zag process runs a continuous process made of segments that turn from one diagonal to the next at random times driven by a generator connected with the components of the gradient of the target log-density. Plus a symmetric term. Provided those random times can be generated, this process is truly available and associated with the right target distribution. When the components of the parameter are independent (an unlikely setting), those random times can be associated with an inhomogeneous Poisson process. In the general case, one needs to bound the gradients by more manageable functions that create a Poisson process that can later be thinned. Next, one needs to simulate the process for the upper bound, a task that seems hard to achieve apart from linear and piecewise constant upper bounds. The process has a bit of a slice sampling taste, except that it cannot be used as a slice sampler but requires continuous time integration, given that the length of each segment matters. (Or maybe random time subsampling?)

A highly innovative part of the paper concentrates on Big Data likelihoods and on the possibility to subsample properly and exactly the original dataset. The authors propose Zig-Zag with subsampling by turning the gradients into random parts of the gradients. While remaining unbiased. There may be a cost associated with this gain of one to *n*, namely that the upper bounds may turn larger as they handle all elements in the likelihood at once, hence become (even) less efficient. (I am more uncertain about the case of the control variates, as it relies on a Lipschitz assumption.) While I still miss an easy way to implement the approach in a specific model, I remain hopeful for this new approach to make a major dent in the current methodologies!