adaptive and delayed MCMC for expensive likelihoods [reply from the authors]

[Chris Sherlock, Andrew Golightly and Daniel Henderson have written a reply about my earlier comments on their arXived paper which works better as a post than as a comment:]

Thank you for the constructive criticism of our paper. Our approach uses a simple weighted average of nearest neighbours and we agree that GPs offer a useful alternative. Both methods have pros and cons, however we first note a similarity: Kriging using a GP also leads to a weighted average of values.

The two most useful pros of the GP are that, (i) by estimating the parameters of the GP one may represent the scales of variability more accurately than a simple nearest neighbour approach with weighting according to Euclidean distance, and (ii) one obtains a distribution for the uncertainty in the Kriging estimate of the log-likelihood.

Both the papers in the blog entry (as well as other recent papers which use GPs), in one way or another take advantage of the second point. However, as acknowledged in Richard Wilkinson’s paper, estimating the parameters of a GP is computationally very costly, and this estimation must be repeated as the training data set grows. Probably for this reason and because of the difficulty in identifying p(p+1)/2 kernel range parameters, Wilkinson’s paper uses a diagonal covariance structure for the kernel. We can find no description of the structure of the covariance function that is used for each statistic in the Meeds & Welling paper but this issue is difficult to avoid.

Our initial training run is used to transform the parameters so that they are approximately orthogonal with unit variance and Euclidean distance is a sensible metric. This has two consequences: (i) the KD-tree is easier to set up and use, and (ii) the nearest neighbours in a KD-tree that is approximately balanced can be found in O(log N) operations, where N is the number of training points. Both (i) and (ii) only require Euclidean distance to be a reasonable measure, not perfect, so there is no need for the training run to have “properly converged”, just for it to represent the gross relationships in the posterior and for the transformation to be 1-1. We note a parallel between our approximate standardisation using training data, and the need to estimate a symmetric matrix of distance parameters from training data to obtain a fully representative GP kernel.

The GP approach might lead to a more accurate estimate of the posterior than a nearest neighbour approach (for a fixed number of training points), but this is necessary for the algorithms in the papers mentioned above since they sample from an approximation to the posterior. As noted in the blog post the delayed-acceptance step (which also could be added to GP-based algorithms) ensures that our algorithm samples from the true posterior so accuracy is helpful for efficiency rather than essential for validity.

We have made the kd-tree C code available and put some effort into making the interface straightforward to use. Our starting point is an existing simple MCMC algorithm; as it is already evaluating the posterior (or an unbiased approximation) then why not store this and take advantage of it within the existing algorithm? We feel that our proposal offers a relatively cheap and straightforward route for this.

adaptive and delayed MCMC for expensive likelihoods

Chris Sherlock, Andrew Golightly and Daniel Henderson recently arXived a paper on a new kind of delayed acceptance.

“With simplicity in mind, we focus on a k-nearest neighbour regression model as the cheap surrogate.”

The central notion in the paper is to extrapolate from values of the likelihoods at a few points in the parameter space towards the whole space through a k-nearest neighbour estimate. While this solution is simple and relatively cheap to compute, it is unclear it is a good surrogate because it does not account for the structure of the model while depending on the choice of a distance. Recent works on Gaussian process approximations seem more relevant. See e.g. papers by Ed Meeds and Max Welling, or by Richard Wilkinson for ABC versions. Obviously, because this is a surrogate only for the first stage delayed acceptance (while the second stage is using the exact likelihood, as in our proposal), the approximation does not have to be super-tight. It should also favour the exploration of tails since (a) any proposal θ outside the current support of the chain is allocated a surrogate value that is the average of its k neighbours, hence larger than the true value in the tails, and (b) due to the delay a larger scale can be used in the random walk proposal. As the authors acknowledge, the knn method deteriorates quickly with the dimension. And computing the approximation grows with the number of MCMC iterations, given that the algorithm is adaptive and uses the exact likelihood values computed so far. Only for the first stage approximation, though, which explains “why” the delayed acceptance algorithm converges. I wondered for a short while whether this was enough to justify convergence, given that the original Metropolis-Hastings probability is just broken into two parts. Since the second stage compensates for the use of a surrogate on the first step, it should not matter in the end. However, the rejection of a proposal still depends on this approximation, i.e., differs from the original algorithm, and hence is turning the Markov chain into a non-Markovian process.

“The analysis sheds light on how computationally cheap the deterministic approximation needs to be to make its use worthwhile and on the relative importance of it matching the `location’ and curvature of the target.”

I had missed the “other” paper by some of the authors on the scaling of delayed acceptance, where they “assume that the error in the cheap deterministic approximation is a realisation of a random function” (p.3).  In which they provide an optimal scaling result for high dimensions à la Roberts et al. (1997), namely a scale of 2.38 (times the target scale) in the random walk proposal. The paper however does not describe the cheap approximation to the target or pseudo-marginal version.

A large chunk of the paper is dedicated to the construction and improvement of the KD-tree used to find the k nearest neighbours. In O(d log(n)) time. Algorithm on which I have no specific comment. Except maybe that the construction of a KD-tree in accordance with a Mahalanobis distance discussed in Section 2.1 requires that the MCMC algorithm has properly converged, which is unrealistic. And also that the construction of a balanced tree seems to require heavy calibrations.

The paper is somewhat harder to read than need be (?) because the authors cumulate the idea of delayed acceptance based on this knn approximation with the technique of pseudo-marginal Metropolis-Hastings. While there is an added value in doing so it complexifies the exposition. And leads to ungainly acronyms like adaptive “da-PsMMH”, which simply are un-readable (!).

I would suggest some material to be published as supplementary material and the overall length of the paper to be reduced. For instance, Section 4.2 is not particularly conclusive. See, e.g., Theorem 2. Or the description of the simulated models in Section 5, which is sometimes redundant.

debunking a (minor and personal) myth

For quite a while, I entertained the idea that Beta and Dirichlet proposals  were more adequate than (log-)normal random walks proposals for parameters on (0,1) and simplicia (simplices, simplexes), respectively, when running an MCMC. For instance, for p in (0,1) the value of the Markov chain at time t-1, the proposal at time t could be a Be(εp,ε{1-p}) generator, since its mean is equal to p and its variance is proportional to 1/(1+ε). (Although I cannot find track of this notion in my books.) The parameter ε can be calibrated towards a given acceptance rate, like the golden number 0.234 of Gelman, Gilks and Roberts (1996). However, when using this proposal on a mixture model, Kaniav Kamari and myself realised today that there is a catch, namely that pushing ε down to achieve an acceptance rate near 0.234 may end up in disaster, since the parameters of the Beta or of the Dirichlet may become lower than 1, which implies an infinite explosion on some boundaries of the parameter space. An explosion that gets more and more serious as ε decreases to zero. Hence is more and more likely to decrease the acceptance rate, thus to reduce ε, which in turns concentrates even more the support on the boundary and leads to a vicious circle and no convergence to the target acceptance rate…

Kamiltonian Monte Carlo [reply]

Heiko Strathmann, Dino Sejdinovic, Samuel Livingstone, Zoltán Szabó, and Arthur Gretton arXived paper about Kamiltonian MCMC generated comments from Michael Betancourt, Dan Simpson and myself, which themselves induced the following reply by Heiko, detailed enough to deserve a post of its own.

Adaptation and ergodicity.
We certainly agree that the naive approach of using a non-parametric kernel density estimator on the chain history (as in [Christian’s book, Example 8.8]) as a *proposal* fails spectacularly on simple examples: the probability of proposing in unexplored regions is extremely small, independent of the current position of the MCMC trajectory. This is not what we do though. Instead, we use the gradient of a density estimator, and not the density itself, for our HMC proposal. Just like KAMH, KMC lite in fact falls back to Random Walk Metropolis in previously unexplored regions and therefore inherits geometric ergodicity properties. This in particular includes the ability to explore previously “unseen” regions, even if adaptation has stopped. I implemented a simple illustration and comparison here.

ABC example.
The main point of the ABC example, is that our method does not suffer from the additional bias from Gaussian synthetic likelihoods when being confronted with skewed models. But there is also a computational efficiency aspect. The scheme by Meeds et al. relies on finite differences and requires $2D$ simulations from the likelihood *every time* the gradient is evaluated (i.e. every leapfrog iteration) and H-ABC discards this valuable information subsequently. In contrast, KMC accumulates gradient information from simulations: it only requires to simulate from the likelihood *once* in the accept/reject step after the leapfrog integration (where gradients are available in closed form). The density is only updated then, and not during the leapfrog integration. Similar work on speeding up HMC via energy surrogates can be applied in the tall data scenario.

Monte Carlo gradients.
Approximating HMC when gradients aren’t available is in general a difficult problem. One approach (like surrogate models) may work well in some scenarios while a different approach (i.e. Monte Carlo) may work better in others, and the ABC example showcases such a case. We very much doubt that one size will fit all — but rather claim that it is of interest to find and document these scenarios.
Michael raised the concern that intractable gradients in the Pseudo-Marginal case can be avoided by running an MCMC chain on the joint space (e.g. $(f,\theta)$ for the GP classifier). To us, however, the situation is not that clear. In many cases, the correlations between variables can cause convergence problems (see e.g. here) for the MCMC and have to be addressed by de-correlation schemes (as here), or e.g. by incorporating geometric information, which also needs fixes as Michaels’s very own one. Which is the method of choice with a particular statistical problem at hand? Which method gives the smallest estimation error (if that is the goal?) for a given problem? Estimation error per time? A thorough comparison of these different classes of algorithms in terms of performance related to problem class would help here. Most papers (including ours) only show experiments favouring their own method.

GP estimator quality.
Finally, to address Michael’s point on the consistency of the GP estimator of the density gradient: this is discussed In the original paper on the infinite dimensional exponential family. As Michael points out, higher dimensional problems are unavoidably harder, however the specific details are rather involved. First, in terms of theory: both the well-specified case (when the natural parameter is in the RKHS, Section 4), and the ill-specified case (the natural parameter is in a “reasonable”, larger class of functions, Section 5), the estimate is consistent. Consistency is obtained in various metrics, including the L² error on gradients. The rates depend on how smooth the natural parameter is (and indeed a poor choice of hyper-parameter will mean slower convergence). The key point, in regards to Michael’s question, is that the smoothness requirement becomes more restrictive as the dimension increases: see Section 4.2, “range space assumption”.
Second, in terms of practice: we have found in experiments that the infinite dimensional exponential family does perform considerably better than a kernel density estimator when the dimension increases (Section 6). In other words, our density estimator can take advantage of smoothness properties of the “true” target density to get good convergence rates. As a practical strategy for hyper-parameter choice, we cross-validate, which works well empirically despite being distasteful to Bayesians. Experiments in the KMC paper also indicate that we can scale these estimators up to dimensions in the 100s on Laptop computers (unlike most other gradient estimation techniques in HMC, e.g. the ones in your HMC & sub-sampling note, or the finite differences in Meeds et al).

 

 

Kamiltonian Monte Carlo [no typo]

Heiko Strathmann, Dino Sejdinovic, Samuel Livingstone, Zoltán Szabó, and Arthur Gretton arXived a paper last week about Kamiltonian MCMC, the K being related with RKHS. (RKHS as in another KAMH paper for adaptive Metropolis-Hastings by essentially the same authors, plus Maria Lomeli and Christophe Andrieu. And another paper by some of the authors on density estimation via infinite exponential family models.) The goal here is to bypass the computation of the derivatives in the moves of the Hamiltonian MCMC algorithm by using a kernel surrogate. While the genuine RKHS approach operates within an infinite exponential family model, two versions are proposed, KMC lite with an increasing sequence of RKHS subspaces, and KMC finite, with a finite dimensional space. In practice, this means using a leapfrog integrator with a different potential function, hence with a different dynamics.

The estimation of the infinite exponential family model is somewhat of an issue, as it is estimated from the past history of the Markov chain, simplified into a random subsample from this history [presumably without replacement, meaning the Markovian structure is lost on the subsample]. This is puzzling because there is dependence on the whole past, which cancels ergodicity guarantees… For instance, we gave an illustration in Introducing Monte Carlo Methods with R [Chapter 8] of the poor impact of approximating the target by non-parametric kernel estimates. I would thus lean towards the requirement of a secondary Markov chain to build this kernel estimate. The authors are obviously aware of this difficulty and advocate an attenuation scheme. There is also the issue of the cost of a kernel estimate, in O(n³) for a subsample of size n. If, instead, a fixed dimension m for the RKHS is selected, the cost is in O(tm²+m³), with the advantage of a feasible on-line update, making it an O(m³) cost in fine. But again the worry of using the whole past of the Markov chain to set its future path…

Among the experiments, a KMC for ABC that follows the recent proposal of Hamiltonian ABC by Meeds et al. The arguments  are interesting albeit sketchy: KMC-ABC does not require simulations at each leapfrog step, is it because the kernel approximation does not get updated at each step? Puzzling.

I also discussed the paper with Michael Betancourt (Warwick) and here his comments:

“I’m hesitant for the same reason I’ve been hesitant about algorithms like Bayesian quadrature and GP emulators in general. Outside of a few dimensions I’m not convinced that GP priors have enough regularization to really specify the interpolation between the available samples, so any algorithm that uses a single interpolation will be fundamentally limited (as I believe is born out in non-trivial scaling examples) and trying to marginalize over interpolations will be too awkward.

They’re really using kernel methods to model the target density which then gives the gradient analytically. RKHS/kernel methods/ Gaussian processes are all the same math — they’re putting prior measures over functions. My hesitancy is that these measures are at once more diffuse than people think (there are lots of functions satisfying a given smoothness criterion) and more rigid than people think (perturb any of the smoothness hyper-parameters and you get an entirely new space of functions).

When using these methods as an emulator you have to set the values of the hyper-parameters which locks in a very singular definition of smoothness and neglects all others. But even within this singular definition there are a huge number of possible functions. So when you only have a few points to constrain the emulation surface, how accurate can you expect the emulator to be between the points?

In most cases where the gradient is unavailable it’s either because (a) people are using decades-old Fortran black boxes that no one understands, in which case there are bigger problems than trying to improve statistical methods or (b) there’s a marginalization, in which case the gradients are given by integrals which can be approximated with more MCMC. Lots of options.”