Archive for reproducing kernel Hilbert space

ABC with kernelised regression

Posted in Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on February 22, 2017 by xi'an

sunset from the Banff Centre, Banff, Canada, March 21, 2012The exact title of the paper by Jovana Metrovic, Dino Sejdinovic, and Yee Whye Teh is DR-ABC: Approximate Bayesian Computation with Kernel-Based Distribution Regression. It appeared last year in the proceedings of ICML.  The idea is to build ABC summaries by way of reproducing kernel Hilbert spaces (RKHS). Regressing such embeddings to the “optimal” choice of summary statistics by kernel ridge regression. With a possibility to derive summary statistics for quantities of interest rather than for the entire parameter vector. The use of RKHS reminds me of Arthur Gretton’s approach to ABC, although I see no mention made of that work in the current paper.

In the RKHS pseudo-linear formulation, the prediction of a parameter value given a sample attached to this value looks like a ridge estimator in classical linear estimation. (I thus wonder at why one would stop at the ridge stage instead of getting the full Bayes treatment!) Things get a bit more involved in the case of parameters (and observations) of interest, as the modelling requires two RKHS, because of the conditioning on the nuisance observations. Or rather three RHKS. Since those involve a maximum mean discrepancy between probability distributions, which define in turn a sort of intrinsic norm, I also wonder at a Wasserstein version of this approach.

What I find hard to understand in the paper is how a large-dimension large-size sample can be managed by such methods with no visible loss of information and no explosion of the computing budget. The authors mention Fourier features, which never rings a bell for me, but I wonder how this operates in a general setting, i.e., outside the iid case. The examples do not seem to go into enough details for me to understand how this massive dimension reduction operates (and they remain at a moderate level in terms of numbers of parameters). I was hoping Jovana Mitrovic could present her work here at the 17w5025 workshop but she sadly could not make it to Banff for lack of funding!

local kernel reduction for ABC

Posted in Books, pictures, Statistics, University life with tags , , , , , on September 14, 2016 by xi'an

“…construction of low dimensional summary statistics can be performed as in a black box…”

Today Zhou and Fukuzumi just arXived a paper that proposes a gradient-based dimension reduction for ABC summary statistics, in the spirit of RKHS kernels as advocated, e.g., by Arthur Gretton. Here the projection is a mere linear projection Bs of the vector of summary statistics, s, where B is an estimated Hessian matrix associated with the posterior expectation E[θ|s]. (There is some connection with the latest version of Li’s and Fearnhead’s paper on ABC convergence as they also define a linear projection of the summary statistics, based on asymptotic arguments, although their matrix does depend on the true value of the parameter.) The linearity sounds like a strong restriction [to me] especially when the summary statistics have no reason to belong to a vectorial space and thus be open to changes of bases and linear projections. For instance, a specific value taken by a summary statistic, like 0 say, may be more relevant than the range of their values. On a larger scale, I am doubtful about always projecting a vector of summary statistics on a subspace with the smallest possible dimension, ie the dimension of θ. In practical settings, it seems impossible to derive the optimal projection and a subvector is almost certain to loose information against a larger vector.

“Another proposal is to use different summary statistics for different parameters.”

Which is exactly what we did in our random forest estimation paper. Using a different forest for each parameter of interest (but no real tree was damaged in the experiment!).

control functionals for Monte Carlo integration

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on June 28, 2016 by xi'an

img_2451A paper on control variates by Chris Oates, Mark Girolami (Warwick) and Nicolas Chopin (CREST) appeared in a recent issue of Series B. I had read and discussed the paper with them previously and the following is a set of comments I wrote at some stage, to be taken with enough gains of salt since Chris, Mark and Nicolas answered them either orally or in the paper. Note also that I already discussed an earlier version, with comments that are not necessarily coherent with the following ones! [Thanks to the busy softshop this week, I resorted to publish some older drafts, so mileage can vary in the coming days.]

First, it took me quite a while to get over the paper, mostly because I have never worked with reproducible kernel Hilbert spaces (RKHS) before. I looked at some proofs in the appendix and at the whole paper but could not spot anything amiss. It is obviously a major step to uncover a manageable method with a rate that is lower than √n. When I set my PhD student Anne Philippe on the approach via Riemann sums, we were quickly hindered by the dimension issue and could not find a way out. In the first versions of the nested sampling approach, John Skilling had also thought he could get higher convergence rates before realising the Monte Carlo error had not disappeared and hence was keeping the rate at the same √n speed.

The core proof in the paper leading to the 7/12 convergence rate relies on a mathematical result of Sun and Wu (2009) that a certain rate of regularisation of the function of interest leads to an average variance of order 1/6. I have no reason to mistrust the result (and anyway did not check the original paper), but I am still puzzled by the fact that it almost immediately leads to the control variate estimator having a smaller order variance (or at least variability). On average or in probability. (I am also uncertain on the possibility to interpret the boxplot figures as establishing super-√n speed.)

Another thing I cannot truly grasp is how the control functional estimator of (7) can be both a mere linear recombination of individual unbiased estimators of the target expectation and an improvement in the variance rate. I acknowledge that the coefficients of the matrices are functions of the sample simulated from the target density but still…

Another source of inner puzzlement is the choice of the kernel in the paper, which seems too simple to be able to cover all problems despite being used in every illustration there. I see the kernel as centred at zero, which means a central location must be know, decreasing to zero away from this centre, so possibly missing aspects of the integrand that are too far away, and isotonic in the reference norm, which also seems to preclude some settings where the integrand is not that compatible with the geometry.

I am equally nonplussed by the existence of a deterministic bound on the error, although it is not completely deterministic, depending on the values of the reproducible kernel at the points of the sample. Does it imply anything restrictive on the function to be integrated?

A side remark about the use of intractable in the paper is that, given the development of a whole new branch of computational statistics handling likelihoods that cannot be computed at all, intractable should possibly be reserved for such higher complexity models.

AISTATS 2016 [#1]

Posted in pictures, R, Running, Statistics, Travel, Wines with tags , , , , , , , , , , , , on May 11, 2016 by xi'an

Travelling through Seville, I arrived in Càdiz on Sunday night, along with a massive depression [weather-speaking!]. Walking through the city from the station was nonetheless pleasant as this is an town full of small streets and nice houses. If with less churches than Seville! Richard Samworth gave the first plenary talk of AISTATS 2016  with a presentation on random projections for classification. His classifier is based on an average of a large number of linear random projections of the original data where the projections are chosen as minimising the prediction error over a subset of the components. The performances of this approach seem to be consistently higher than for random forests, which makes it definitely worth investigating further. (A related R package is available.)

The following talks that day covered Bayesian optimisation and probabilistic numerics, with Javier Gonzales introducing glasses for Bayesian optimisation in order to solve its myopia (!)—by which he meant predicting the output of the optimisation over n future steps. And a first mention of the Pima Indians by Daniel Hernandez-Lobato in his talk about EP with stochastic gradient steps towards optimisation. (As well as much larger datasets.) And Mark Girolami bringing quasi-Monte Carlo into control variates. A kernel based ABC by Mijung Park, which uses kernels and maximum mean discrepancy to avoid defining summary statistics, and a version of parallel MCMC by Guillaume Basse. Plus another session on deep learning.

As usual with AISTATS conferences, the central activity of the day was the noon poster session, including speakers discussing their paper, and I had several interesting chats about MCMC related topics, with e.g. one alternative notion of ensemble MCMC [centred on estimating the normalising constant].

We awarded the notable student paper awards before the welcoming cocktail: The winners are Bo DaiNedelina Teneva, and Ye Wang.  And this first day ended up with a companionable evening in a most genuine tapa bar, tasting local blood sausage and local blue cheese. (If you do not mind the corrida theme!)

Kamiltonian Monte Carlo [reply]

Posted in Books, Statistics, University life with tags , , , , , , , , , , on July 3, 2015 by xi'an

kamilHeiko 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]

Posted in Books, Statistics, University life with tags , , , , , , , , , , on June 29, 2015 by xi'an

kamilHeiko 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.”

Quasi-Monte Carlo sampling

Posted in Books, Kids, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , on December 10, 2014 by xi'an

RSS wine“The QMC algorithm forces us to write any simulation as an explicit function of uniform samples.” (p.8)

As posted a few days ago, Mathieu Gerber and Nicolas Chopin will read this afternoon a Paper to the Royal Statistical Society on their sequential quasi-Monte Carlo sampling paper.  Here are some comments on the paper that are preliminaries to my written discussion (to be sent before the slightly awkward deadline of Jan 2, 2015).

Quasi-Monte Carlo methods are definitely not popular within the (mainstream) statistical community, despite regular attempts by respected researchers like Art Owen and Pierre L’Écuyer to induce more use of those methods. It is thus to be hoped that the current attempt will be more successful, it being Read to the Royal Statistical Society being a major step towards a wide diffusion. I am looking forward to the collection of discussions that will result from the incoming afternoon (and bemoan once again having to miss it!).

“It is also the resampling step that makes the introduction of QMC into SMC sampling non-trivial.” (p.3)

At a mathematical level, the fact that randomised low discrepancy sequences produce both unbiased estimators and error rates of order

\mathfrak{O}(N^{-1}\log(N)^{d-}) \text{ at cost } \mathfrak{O}(N\log(N))

means that randomised quasi-Monte Carlo methods should always be used, instead of regular Monte Carlo methods! So why is it not always used?! The difficulty stands [I think] in expressing the Monte Carlo estimators in terms of a deterministic function of a fixed number of uniforms (and possibly of past simulated values). At least this is why I never attempted at crossing the Rubicon into the quasi-Monte Carlo realm… And maybe also why the step had to appear in connection with particle filters, which can be seen as dynamic importance sampling methods and hence enjoy a local iid-ness that relates better to quasi-Monte Carlo integrators than single-chain MCMC algorithms.  For instance, each resampling step in a particle filter consists in a repeated multinomial generation, hence should have been turned into quasi-Monte Carlo ages ago. (However, rather than the basic solution drafted in Table 2, lower variance solutions like systematic and residual sampling have been proposed in the particle literature and I wonder if any of these is a special form of quasi-Monte Carlo.) In the present setting, the authors move further and apply quasi-Monte Carlo to the particles themselves. However, they still assume the deterministic transform

\mathbf{x}_t^n = \Gamma_t(\mathbf{x}_{t-1}^n,\mathbf{u}_{t}^n)

which the q-block on which I stumbled each time I contemplated quasi-Monte Carlo… So the fundamental difficulty with the whole proposal is that the generation from the Markov proposal

m_t(\tilde{\mathbf{x}}_{t-1}^n,\cdot)

has to be of the above form. Is the strength of this assumption discussed anywhere in the paper? All baseline distributions there are normal. And in the case it does not easily apply, what would the gain bw in only using the second step (i.e., quasi-Monte Carlo-ing the multinomial simulation from the empirical cdf)? In a sequential setting with unknown parameters θ, the transform is modified each time θ is modified and I wonder at the impact on computing cost if the inverse cdf is not available analytically. And I presume simulating the θ’s cannot benefit from quasi-Monte Carlo improvements.

The paper obviously cannot get into every detail, obviously, but I would also welcome indications on the cost of deriving the Hilbert curve, in particular in connection with the dimension d as it has to separate all of the N particles, and on the stopping rule on m that means only Hm is used.

Another question stands with the multiplicity of low discrepancy sequences and their impact on the overall convergence. If Art Owen’s (1997) nested scrambling leads to the best rate, as implied by Theorem 7, why should we ever consider another choice?

In connection with Lemma 1 and the sequential quasi-Monte Carlo approximation of the evidence, I wonder at any possible Rao-Blackwellisation using all proposed moves rather than only those accepted. I mean, from a quasi-Monte Carlo viewpoint, is Rao-Blackwellisation easier and is it of any significant interest?

What are the computing costs and gains for forward and backward sampling? They are not discussed there. I also fail to understand the trick at the end of 4.2.1, using SQMC on a single vector instead of (t+1) of them. Again assuming inverse cdfs are available? Any connection with the Polson et al.’s particle learning literature?

Last questions: what is the (learning) effort for lazy me to move to SQMC? Any hope of stepping outside particle filtering?