Archive for Bayesian GANs

mean field Langevin system & neural networks

Posted in Statistics with tags , , , , , , , on February 4, 2020 by xi'an

A colleague of mine in Paris Dauphine, Zhenjie Ren, recently gave a talk on recent papers of his connecting neural nets and Langevin. Estimating the parameters of the NNs by mean-field Langevin dynamics. Following from an earlier paper on the topic by Mei, Montanari & Nguyen in 2018. Here are some notes I took during the seminar, not necessarily coherent as I was a bit under the weather that day. And had no previous exposure to most notions.

Fitting a one-layer network is turned into a minimisation programme over a measure space (when using loads of data). A reformulation that makes the problem convex. Adding a regularisation by the entropy and introducing derivatives of a functional against the measure. With a necessary and sufficient condition for the solution to be unique when the functional is convex. This reformulation leads to a Fokker-Planck equation, itself related with a Langevin diffusion. Except there is a measure in the Langevin equation, which stationary version is the solution of the original regularised minimisation programme.

A second paper contains an extension to deep NN, re-expressed as a problem in a random environment. Or with a marginal constraint (one marginal distribution being constrained). With a partial derivative wrt the marginal measure. Turning into a Langevin diffusion with an extra random element. Using optimal control produces a new Hamiltonian. Eventually producing the mean-field Langevin system as backward propagation. Coefficients being computed by chain rule, equivalent to a Euler scheme for Langevin dynamics.

This approach holds consequence for GANs with discriminator as one-layer NN and generator minimised over two measures. The discriminator is the invariant measure of the mean-field Langevin dynamics. Mentioning Metropolis-Hastings GANs which seem to require one full run of an MCMC algorithm at each iteration of the mean-field Langevin.

Bayesian inference with no likelihood

Posted in Books, Statistics, University life with tags , , , , , , , , on January 28, 2020 by xi'an

This week I made a quick trip to Warwick for the defence (or viva) of the PhD thesis of Jack Jewson, containing novel perspectives on constructing Bayesian inference without likelihood or without complete trust in said likelihood. The thesis aimed at constructing minimum divergence posteriors in an M-open perspective and built a rather coherent framework from principles to implementation. There is a clear link with the earlier work of Bissiri et al. (2016), with further consistency constraints where the outcome must recover the true posterior in the M-closed scenario (if not always the case with the procedures proposed in the thesis).

Although I am partial to the use of empirical likelihoods in setting, I appreciated the position of the thesis and the discussion of the various divergences towards the posterior derivation (already discussed on this blog) , with interesting perspectives on the calibration of the pseudo-posterior à la Bissiri et al. (2016). Among other things, the thesis pointed out a departure from the likelihood principle and some of its most established consequences, like Bayesian additivity. In that regard, there were connections with generative adversarial networks (GANs) and their Bayesian versions that could have been explored. And an impression that the type of Bayesian robustness explored in the thesis has more to do with outliers than with misspecification. Epsilon-contamination amodels re quite specific as it happens, in terms of tails and other things.

The next chapter is somewhat “less” Bayesian in my view as it considers a generalised form of variational inference. I agree that the view of the posterior as a solution to an optimisation is tempting but changing the objective function makes the notion less precise.  Which makes reading it somewhat delicate as it seems to dilute the meaning of both prior and posterior to the point of becoming irrelevant.

The last chapter on change-point models is quite alluring in that it capitalises on the previous developments to analyse a fairly realistic if traditional problem, applied to traffic in London, prior and posterior to the congestion tax. However, there is always an issue with robustness and outliers in that the notion is somewhat vague or informal. Things start clarifying at the end but I find surprising that conjugates are robust optimal solutions since the usual folk theorem from the 80’s is that they are not robust.

an independent sampler that maximizes the acceptance rate of the MH algorithm

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , on September 3, 2019 by xi'an

An ICLR 2019 paper by Neklyudov, Egorov and Vetrov on an optimal choice of the proposal in an independent Metropolis algorithm I discovered via an X validated question. Namely whether or not the expected Metropolis-Hastings acceptance ratio is always one (which it is not when the support of the proposal is restricted). The paper mentions the domination of the Accept-Reject algorithm by the associated independent Metropolis-Hastings algorithm, which has actually been stated in our Monte Carlo Statistical Methods (1999, Lemma 6.3.2) and may prove even older. The authors also note that the expected acceptance probability is equal to one minus the total variation distance between the joint defined as target x Metropolis-Hastings proposal distribution and its time-reversed version. Which seems to suffer from the same difficulty as the one mentioned in the X validated question. Namely that it only holds when the support of the Metropolis-Hastings proposal is at least the support of the target (or else when the support of the joint defined as target x Metropolis-Hastings proposal distribution is somewhat symmetric. Replacing total variation with Kullback-Leibler then leads to a manageable optimisation target if the proposal is a parameterised independent distribution. With a GAN version when the proposal is not explicitly available. I find it rather strange that one still seeks independent proposals for running Metropolis-Hastings algorithms as the result will depend on the family of proposals considered and as performances will deteriorate with dimension (the authors mention a 10% acceptance rate, which sounds quite low). [As an aside, ICLR 2020 will take part in Addis Abeba next April.]

noise contrastive estimation

Posted in Statistics with tags , , , , , , , , , on July 15, 2019 by xi'an

As I was attending Lionel Riou-Durand’s PhD thesis defence in ENSAE-CREST last week, I had a look at his papers (!). The 2018 noise contrastive paper is written with Nicolas Chopin (both authors share the CREST affiliation with me). Which compares Charlie Geyer’s 1994 bypassing the intractable normalising constant problem by virtue of an artificial logit model with additional simulated data from another distribution ψ.

“Geyer (1994) established the asymptotic properties of the MC-MLE estimates under general conditions; in particular that the x’s are realisations of an ergodic process. This is remarkable, given that most of the theory on M-estimation (i.e.estimation obtained by maximising functions) is restricted to iid data.”

Michael Guttman and Aapo Hyvärinen also use additional simulated data in another likelihood of a logistic classifier, called noise contrastive estimation. Both methods replace the unknown ratio of normalising constants with an unbiased estimate based on the additional simulated data. The major and impressive result in this paper [now published in the Electronic Journal of Statistics] is that the noise contrastive estimation approach always enjoys a smaller variance than Geyer’s solution, at an equivalent computational cost when the actual data observations are iid. And the artificial data simulations ergodic. The difference between both estimators is however negligible against the Monte Carlo error (Theorem 2).

This may be a rather naïve question, but I wonder at the choice of the alternative distribution ψ. With a vague notion that it could be optimised in a GANs perspective. A side result of interest in the paper is to provide a minimal (re)parameterisation of the truncated multivariate Gaussian distribution, if only as an exercise for future exams. Truncated multivariate Gaussian for which the normalising constant is of course unknown.

“more Bayesian” GANs

Posted in Books, Statistics with tags , , , , on December 21, 2018 by xi'an
On X validated, I got pointed to this recent paper by He, Wang, Lee and Tiang, that proposes a new form of Bayesian GAN. Although I do not see it as really Bayesian, as explained below.
“[The] existing Bayesian method (Saatchi & Wilson, 2017) may lead to incompatible conditionals, which suggest that the underlying joint distribution actually does not exist.”
The difference with the Bayesian GANs of Saatchi & Wilson (2017) [with Saatchi’s name being consistently misspelled throughout] is in the definition of the likelihood function, function of both generative and discriminatory parameters. As in Bissiri et al. (2013), the likelihood is replaced by the exponentiated loss function, or rather functions, which are computed with expected or pluggin distributions or discriminating functions. Expectations under the respective priors and for the observed data (?). Meaning there are “two likelihoods” for the same data, one being the inverse of the other in the minimax GAN case. Further, the prior on the generative parameter is actually of the prior feedback category:  at each iteration, the authors “use the generator distribution in the previous time step as a prior for the next time step”. Which makes me wonder how they avoid ending up with a Dirac “prior”. (Even curiouser, the prior on the discriminating parameter, which is not a genuine component of the statistical model, is a flat prior across iterations.) The convergence result established in the paper is that, if the (true) data-generating model can be written as the marginal of the assumed parametric generative model against an “optimal” distribution, then the discriminating function converges to non-discrimination between (true) data-generating model and the assumed parametric generative model. This somehow negates the Bayesian side of the affair, as convergence to a point mass does not produce a Bayesian outcome on the parameters of the model, or on the comparison between true and assumed models. The paper also demonstrates the incompatibility of the two conditionals used by Saatchi & Wilson (2017) and provides a formal example [missing any form of data?] where the associated Bayesian GAN does not converge to the true value behind the model. But the issue is more complex in my opinion in that using two incompatible conditionals does not mean that the associated Markov chain is transient (as e.g. on a compact space).

Gibbs for incompatible kids

Posted in Books, Statistics, University life with tags , , , , , , , , , , on September 27, 2018 by xi'an

In continuation of my earlier post on Bayesian GANs, which resort to strongly incompatible conditionals, I read a 2015 paper of Chen and Ip that I had missed. (Published in the Journal of Statistical Computation and Simulation which I first confused with JCGS and which I do not know at all. Actually, when looking at its editorial board,  I recognised only one name.) But the study therein is quite disappointing and not helping as it considers Markov chains on finite state spaces, meaning that the transition distributions are matrices, meaning also that convergence is ensured if these matrices have no null probability term. And while the paper is motivated by realistic situations where incompatible conditionals can reasonably appear, the paper only produces illustrations on two and three states Markov chains. Not that helpful, in the end… The game is still afoot!

Bayesian GANs [#2]

Posted in Books, pictures, R, Statistics with tags , , , , , , , , , , , , on June 27, 2018 by xi'an

As an illustration of the lack of convergence of the Gibbs sampler applied to the two “conditionals” defined in the Bayesian GANs paper discussed yesterday, I took the simplest possible example of a Normal mean generative model (one parameter) with a logistic discriminator (one parameter) and implemented the scheme (during an ISBA 2018 session). With flat priors on both parameters. And a Normal random walk as Metropolis-Hastings proposal. As expected, since there is no stationary distribution associated with the Markov chain, simulated chains do not exhibit a stationary pattern,

And they eventually reach an overflow error or a trapping state as the log-likelihood gets approximately to zero (red curve).

Too bad I missed the talk by Shakir Mohammed yesterday, being stuck on the Edinburgh by-pass at rush hour!, as I would have loved to hear his views about this rather essential issue…