training energy based models

This 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.

JB³ [Junior Bayes beyond the borders]

Bocconi and j-ISBA are launcing a webinar series for and by junior Bayesian researchers. The first talk is on 25 June, 25 at 3pm UTC/GMT (5pm CET) with Francois-Xavier Briol, one of the laureates of the 2020 Savage Thesis Prize (and a former graduate of OxWaSP, the Oxford-Warwick doctoral training program), on Stein’s method for Bayesian computation, with as a discussant Nicolas Chopin.

As pointed out on their webpage,

Due to the importance of the above endeavor, JB³ will continue after the health emergency as an annual series. It will include various refinements aimed at increasing the involvement of the whole junior Bayesian community and facilitating a broader participation to the online seminars all over the world via various online solutions.

Thanks to all my friends at Bocconi for running this experiment!

is there such a thing as optimal subsampling?

This idea of optimal thinnin and burnin has been around since the early days of the MCMC revolution and did not come up with a definite answer. For instance, from a pure estimation perspective, subsampling always increases the variance of the resulting estimator. My personal approach is to ignore both burnin and thinnin and rather waste time on running several copies of the code to check for potential discrepancies and get a crude notion of the variability. And to refuse to answer to questions like is 5000 iterations long enough for burnin?

A recent arXival by Riabiz et al. readdresses the issue. In particular concerning this notion that the variance of the subsampled version is higher: this only applies to a deterministic subsampling, as opposed to an MCMC-based subsampling (although this intricacy only makes the problem harder!). I however fail to understand the argument in favour of subsampling based on storage issues (p.4), as a dynamic storage of the running mean for all quantities of interest does not cost anything if the integrand is not particularly demanding. I also disagree at the pessimistic view that the asymptotic variance of the MCMC estimate is hard to estimate: papers by Flegal, Hobert, Jones, Vat and others have rather clearly shown how batch means can produce converging estimates of this asymptotic variance.

“We do not to attempt to solve a continuous optimisation problem for selection of the next point [in the sample]. Such optimisation problems are fundamentally difficult and can at best be approximately solved. Instead, we exactly solve the discrete optimisation problem of selecting a suitable element from a supplied MCMC output.”

One definitely positive aspect of the paper is that the (thinning) method is called Stein thinning, in connection with Stein’s discrepancy, and this honours Charles Stein. The method looks at the optimal subsample, with optimality defined in terms of minimising Stein’s discrepancy from the true target over a reproducible kernel Hilbert space. And then over a subsample to minimise the distance from the empirical distribution to the theoretical distribution. The kernel (11) is based on the gradient of the target log density and the solution is determined by greedy algorithms that determine which next entry to add to the empirical distribution. Which is of complexity O(nm²) if the subsample is of size m. Some entries may appear more than once and the burnin step could be automatically included as (relatively) unlikely values are never selected (at least this was my heuristic understanding). While the theoretical backup for the construct is present and backed by earlier papers of some of the authors, I do wonder at the use of the most rudimentary representation of an approximation to the target when smoother versions could have been chosen and optimised on the same ground. And I am also surprised at the dependence of both estimators and discrepancies on the choice of the (sort-of) covariance matrix in the inner kernel, as the ODE examples provided in the paper (see, e.g., Figure 7). (As an aside and at a shallow level, the approach also reminded me of the principal points of my late friend Bernhard Flury…) Storage of all MCMC simulations for a later post-processing is of course costly in terms of storage, at O(nm). Unless a “secretary problem” approach can be proposed to get sequential. Another possible alternate would be to consider directly the chain of the accepted values (à la vanilla Rao-Blackwellisation). Overall, since the stopping criterion is based on a fixed sample size, and hence depends on the sub-efficiency of evaluating the mass of different modes, I am unsure the method is anything but what-you-get-is-what-you-see, i.e. prone to get misled by a poor exploration of the complete support of the target.

“This paper focuses on nonuniform subsampling and shows that it is more efficiency than uniform subsampling.”

Two weeks later, Guanyu Hu and Hai Ying Wang arXived their Most Likely Optimal Subsampled Markov Chain Monte Carlo, in what I first thought as an answer to the above! But both actually have little in common as this second paper considers subsampling on the data, rather than the MCMC output, towards producing scalable algorithms. Building upon Bardenet et al. (2014) and Korattikara et al. (2014).  Replacing thus the log-likelihood with a random sub-sampled version and deriving the sample size based on a large deviation inequality. By a Cauchy-Schwartz inequality, the authors find sampling probabilities proportional to the individual log-likelihooods. Which depend on the running value of the MCMC’ed parameters. And thus replaced with the values at a fixed parameter, with cost O(n) but only once, but no so much optimal. (The large deviation inequality therein is only concerned with an approximation to the log-likelihood, without examining the long term impact on the convergence of the approximate Markov chain as this is no longer pseudo-marginal MCMC. For instance, both current and prospective log-likelihoods are re-estimated at each iteration. The paper compares with uniform sampling on toy examples,  to demonstrate a smaller estimation error for the statistical problem, rather than convergence to the true posterior.)

Stein’s method in machine learning [workshop]

There will be an ICML workshop on Stein’s method in machine learning & statistics, next July 14 or 15, located in Long Beach, CA. Organised by François-Xavier Briol (formerly Warwick), Lester Mckey, Chris Oates (formerly Warwick), Qiang Liu, and Larry Golstein. To quote from the webpage of the workshop

Stein’s method is a technique from probability theory for bounding the distance between probability measures using differential and difference operators. Although the method was initially designed as a technique for proving central limit theorems, it has recently caught the attention of the machine learning (ML) community and has been used for a variety of practical tasks. Recent applications include goodness-of-fit testing, generative modeling, global non-convex optimisation, variational inference, de novo sampling, constructing powerful control variates for Monte Carlo variance reduction, and measuring the quality of Markov chain Monte Carlo algorithms.

Speakers include Anima Anandkumar, Lawrence Carin, Louis Chen, Andrew Duncan, Arthur Gretton, and Susan Holmes. I am quite sorry to miss two workshops dedicated to Stein’s work in a row, the other one being at NUS, Singapore, around the Stein paradox.

“Stein deviates from the statistical norm”