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…
Archive for noise contrasting estimation
[more than] everything you always wanted to know about marginal likelihood
Posted in Books, Statistics, University life with tags adaptive importance sampling, arXiv, Bayes factor, bridge sampling, candidate's formula, Charlie Geyer, Chib's approximation, CRiSM, improper prior, Julian Besag, Laplace approximation, Madrid, nested sampling, noise contrasting estimation, path sampling, reversible jump MCMC, sequential Monte Carlo, surveys, umbrella sampling, Universidad Carlos III de Madrid, University of Warwick, Warwickshire on February 10, 2022 by xi'an
Earlier this year, F. Llorente, L. Martino, D. Delgado, and J. Lopez-Santiago have arXived an updated version of their massive survey on marginal likelihood computation. Which I can only warmly recommend to anyone interested in the matter! Or looking for a base camp to initiate a graduate project. They break the methods into four families
- Deterministic approximations (e.g., Laplace approximations)
- Methods based on density estimation (e.g., Chib’s method, aka the candidate’s formula)
- Importance sampling, including sequential Monte Carlo, with a subsection connecting with MCMC
- Vertical representations (mostly, nested sampling)
Besides sheer computation, the survey also broaches upon issues like improper priors and alternatives to Bayes factors. The parts I would have done in more details are reversible jump MCMC and the long-lasting impact of Geyer’s reverse logistic regression (with the noise contrasting extension), even though the link with bridge sampling is briefly mentioned there. There is even a table reporting on the coverage of earlier surveys. Of course, the following postnote of the manuscript
The Christian Robert’s blog deserves a special mention , since Professor C. Robert has devoted several entries of his blog with very interesting comments regarding the marginal likelihood estimation and related topics.
does not in the least make me less objective! Some of the final recommendations
- use of Naive Monte Carlo [simulate from the prior] should be always considered [assuming a proper prior!]
- a multiple-try method is a good choice within the MCMC schemes
- optimal umbrella sampling estimator is difficult and costly to implement , so its best performance may not be achieved in practice
- adaptive importance sampling uses the posterior samples to build a suitable normalized proposal, so it benefits from localizing samples in regions of high posterior probability while preserving the properties of standard importance sampling
- Chib’s method is a good alternative, that provide very good performances [but is not always available]
- the success [of nested sampling] in the literature is surprising.
improving bridge samplers by GANs
Posted in Books, pictures, Statistics with tags bridge sampling, curse of dimensionality, GANs, noise contrasting estimation, normalising flow, PhD, Saint Giles cemetery, University of Oxford on July 20, 2021 by xi'an
Hanwen Xing from Oxford recently posted a paper on arXiv about using GANs to improve the overlap bewtween the densities in bridge sampling. Bringing out new connections with noise contrastive estimation. The idea is to optimise a transform of one of the densities h() to bring it closer to the other density k(), using for instance normalising flows. (The call to transforms for bridge is not new, dating at least to Voter in 1985, the year I was starting my PhD!) Furthermore, using an f-divergence as a measure of functional distance allows for a reasonably straightforward update of the transform. That can be reformulated as a GAN target, which is somewhat natural in that the transform aims at confusing simulation from the transform of h and from k. This is quite an interesting proposal, even though calculating the optimal transform is time-consuming and subjet to the curse of dimensionality. I also wonder at whether or not iterating the optimisation, one density after the other, would be bring further improvement.
NCE, VAEs, GANs & even ABC…
Posted in Statistics with tags ABC, Bayesian GANs, CDT, deep learning, energy based model, generative adversarial networks, noise contrasting estimation, normalising constant, normalising flow, partition function, PhD course, Teams, University of Warwick, variational autoencoders on May 14, 2021 by xi'anAs I was preparing my (new) lectures for a PhD short course “at” Warwick (meaning on Teams!), I read a few surveys and other papers on all these acronyms. It included the massive Guttmann and Hyvärinen 2012 NCE JMLR paper, Goodfellow’s NIPS 2016 tutorial on GANs, and Kingma and Welling 2019 introduction to VAEs. Which I found a wee bit on the light side, maybe missing the fundamentals of the notion… As well as the pretty helpful 2019 survey on normalising flows by Papamakarios et al., although missing on the (statistical) density estimation side. And also a nice (2017) survey of GANs by Shakir Mohamed and Balaji Lakshminarayanan with a somewhat statistical spirit, even though convergence issues are not again not covered. But misspecification is there. And the many connections between ABC and GANs, if definitely missing on the uncertainty aspects. While Deep Learning by Goodfellow, Bengio and Courville adresses both the normalising constant (or partition function) and GANs, it was somehow not deep enough (!) to use for the course, offering only a few pages on NCE, VAEs and GANs. (And also missing on the statistical references addressing the issue, incl. [or excl.] Geyer, 1994.) Overall, the infinite variations offered on GANs leave me uncertain about their statistical relevance, as it is unclear how good the regularisation therein is for handling overfitting and consistent estimation. (And if I spot another decomposition of the Kullback-Leibler divergence, I may start crying…)
training energy based models
Posted in Books, Statistics with tags Charles Stein, energy based model, GAN, generative adversarial networks, Hyvärinen score, maximum likelihood estimation, noise contrasting estimation, unbiased estimation on April 7, 2021 by xi'anThis recent arXival by Song and Kingma covers different computational approaches to semi-parametric estimation, but also exposes imho the chasm existing between statistical and machine learning perspectives on the problem.
“Energy-based models are much less restrictive in functional form: instead of specifying a normalized probability, they only specify the unnormalized negative log-probability (…) Since the energy function does not need to integrate to one, it can be parameterized with any nonlinear regression function.”
The above in the introduction appears first as a strange argument, since the mass one constraint is the least of the problems when addressing non-parametric density estimation. Problems like the convergence, the speed of convergence, the computational cost and the overall integrability of the estimator. It seems however that the restriction or lack thereof is to be understood as the ability to use much more elaborate forms of densities, which are then black-boxes whose components have little relevance… When using such mega-over-parameterised representations of densities, such as neural networks and normalising flows, a statistical assessment leads to highly challenging questions. But convergence (in the sample size) does not appear to be a concern for the paper. (Except for a citation of Hyvärinen on p.5.)
Using MLE in this context appears to be questionable, though, since the base parameter θ is not unlikely to remain identifiable. Computing the MLE is therefore a minor issue, in this regard, a resolution based on simulated gradients being well-chartered from the earlier era of stochastic optimisation as in Robbins & Monro (1954), Duflo (1996) or Benveniste & al. (1990). (The log-gradient of the normalising constant being estimated by the opposite of the gradient of the energy at a random point.)
“Running MCMC till convergence to obtain a sample x∼p(x) can be computationally expensive.”
Contrastive divergence à la Hinton (2002) is presented as a solution to the convergence problem by stopping early, which seems reasonable given the random gradient is mostly noise. With a possible correction for bias à la Jacob & al. (missing the published version).
An alternative to MLE is the 2005 Hyvärinen score, notorious for bypassing the normalising constant. But blamed in the paper for being costly in the dimension d of the variate x, due to the second derivative matrix. Which can be avoided by using Stein’s unbiased estimator of the risk (yay!) if using randomized data. And surprisingly linked with contrastive divergence as well, if a Taylor expansion is good enough an approximation! An interesting byproduct of the discussion on score matching is to turn it into an unintended form of ABC!
“Many methods have been proposed to automatically tune the noise distribution, such as Adversarial Contrastive Estimation (Bose et al., 2018), Conditional NCE (Ceylan and Gutmann, 2018) and Flow Contrastive Estimation (Gao et al., 2020).”
A third approach is the noise contrastive estimation method of Gutmann & Hyvärinen (2010) that connects with both others. And is a precursor of GAN methods, mentioned at the end of the paper via a (sort of) variational inequality.
