Category Archives: stats

Overdispersed Black-Box Variational Inference

This UAI paper by Ruiz, Titsias and Blei presents important insights for the idea of a black box procedure for VI (which I discussed here). The setup of BBVI is the following: given a target/posterior \pi and a parametric approximation q_\lambda, we want to find

\mathrm{argmin}_\lambda \int \log \left ( \frac{\pi(x)}{q_\lambda(x)} \right )  q_\lambda(x) \mathrm{d}x

which can be achieved for any q_\lambda by estimating the gradient

\nabla_\lambda \int \log \left ( \frac{\pi(x)}{q_\lambda(x)} \right )  q_\lambda(x) \mathrm{d}x

with Monte Carlo Samples and stochastic gradient descent. This works if we can easily sample from q_\lambda  and can compute its derivative wrt \lambda in closed form. In the original paper, the authors suggested the use of the score function as a control variate and a Rao-Blackwellization. Both where described in a way that utterly confused me – until now, because Ruiz, Titsias and Blei manage to describe the concrete application of both control variates and Rao-Blackwellization in a very transparent way. Their own contribution to variance reduction (minus some tricks they applied) is based on the fact that the optimal sampling distribution for estimating \nabla_\lambda \int \log \left ( \frac{\pi(x)}{q_\lambda(x)} \right )  q_\lambda(x) \mathrm{d}x is proportional to \left | \log \left ( \frac{\pi(x)}{q_\lambda(x)} \right ) \right |  q_\lambda(x) rather than exactly q_\lambda(x). They argue that this optimal sampling distribution is considerably heavier tailed than q_\lambda(x). Their reasoning is mainly that the norm of the gradient (which is essentially (\nabla_\lambda q_\lambda) \log \left ( \frac{\pi(x)}{q_\lambda(x)} \right )  = q_\lambda(x)(\nabla_\lambda \log q_\lambda(x)) \log \left ( \frac{\pi(x)}{q_\lambda(x)} \right ) )  vanishes for the modes, making that region irrelevant for gradient estimation. The same should be true for the tails of the distribution I think. Overall very interesting work that I strongly recommend reading, if only to understand the original Blackbox VI proposal.

A Variational Analysis of Stochastic Gradient Algorithms

This arXival by Mandt, Hoffman and Blei from February takes a look at what they call SGD with constant learning rate or constant SGD. Which is not really Stochastic Gradient Descent anymore, but rather a sampling algorithm, as we should bounce  around the mode of the posterior rather than converging to it. Consequently they interpret constant SGD as a stochastic process in discrete time with some stationary distribution. They go on to analyse it under the assumption that the stationary distribution is Gaussian (Assumption 1 and 4). Something that springs to my mind here is the following: even if we accept that the stationary distribution of the process might be Gaussian (intuitively, not rigorously), what guarantees that the stationary distribution is the posterior? Even if the posterior is also gaussian, it could have a different (co)variance. I think a few more words about this question might be good. In particular, without adding artificial noise, is it guaranteed that every point in the posteriors domain can be reached with non-zero probability? And can be reached an infinite number of times if we collect infinite samples? If not, the constant SGD process is transient and does not enjoy good stability properties.

Another delicate point hides in Assumption 1: As our stochastic gradient is the sum of independent RVs, the CLT is invoked to assume that the gradient noise is Gaussian. But the CLT might not yet take effect if the number of data points in our subsample is small, which is the very thing one would like in order to have advantages in computation time. This of course is yet another instance of the dilemma common to all scalable sampling techniques, and at this stage of research this is not a show stopper.
Now assuming that constant SGD is not transient and it has a stationary distribution that is the posterior and that the posterior or its dominant mode is approximately Gaussian, we can of course try to optimize the parameters of constant SGD to make the sample it produces as close a possible to a posterior sample. Which the authors conveniently do using an Ornstein-Uhlenbeck approximation, which has a Gaussian stationary distribution. Their approach is to minimize KL-Divergence between the OU approximation and the posterior, and derive optimal parameter settings for constant SGD in closed form. The most interesting part of the theoretical results section probably is the one on optimal preconditioning in Stochastic Gradient Fisher Scoring, as that sampler adds artificial noise. Which might solve some of the transience questions.

The presentation would gain a lot by renaming “constant SGD” for what it is – a sampling algorithm. The terminology and notation in general are sometimes a bit confusing, but nothing a revision can’t fix. In general, the paper presents an interesting approach to deriving optimal parameters for a class of algorithms that is notoriously hard to tune. This is particularly relevant because the constant step size could improve mixing considerably over Welling & Tehs SGLD. What would be interesting to see is wether the results could be generalized to the case where the posterior is not Gaussian. For practical reasons, because stochastic VB/EP works quite well in the gaussian case. For theoretical reasons, because EP now even comes with some guarantees (haven’t read that paper admittedly). Maybe a path would be to take a look at the Gaussian approximation to a multimodal posterior spanning all modes, and minimizing KL-Divergence between that and the OU process. Or maybe one can proof that constant SGD (with artificial noise?) has some fixed stationary distribution to which stochastic drift term plus noise are a Markov Kernel, which might enable a pseudo-marginal correction.

Variational Hamiltonian Monte Carlo via Score Matching

This is an arXival (maybe by now ICML paper?) by Zhang, Shahbaba and Zhao. It suggest fitting an emulator/surrogate q_\eta to the target density \pi via score matching and then use the gradients of q_\eta rather than those of \pi for generating proposal with HMC. Their main selling point being that this decreases wall-clock time for HMC.

However, that seems to me like reselling the earlier paper by Heiko Strathmann and collaborators on Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families. Zhang et al use basically the same idea but fail to look at this earlier work. Their reasoning to use a neural network rather than a GP emulator (computation time) is a bit arbitrary. If you go for a less rich function class (neural network) then the computation time will go down of course – but you would get the same effect by using GPs with inducing points.

Very much lacking to my taste is reasoning for doing the tuning they do. Sometimes they tune HMC/Variational HMC to 85% acceptance, sometimes to 70% acceptance. Also, it seems they not adapting the mass matrix of HMC. If they would, I conjecture the relative efficiency of Standard HMC vs Variational HMC could change drastically. Details on how they tune SGLD is completely lacking.

Overall, I think it is not yet clear what can be learned from the work reported in the paper.

 

 

Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm

This is an important arXival by Alain Durmus and Eric Moulines. The title is slightly optimized for effect, as the paper actually contains non-asymptotic and asymptotic analysis.

The basic theme of the paper is in getting upper bounds on total variation (and more general distribution distances) between an uncorrected discretized Langevin diffusion wrt some target \pi and \pi itself. The discretization used is the common scheme with the scary name Euler-Maruyama:

X_{k} \sim \mathcal{N}(\cdot|X_{k-1} + \gamma_{k}\nabla \log \pi(X_{k-1}), \sqrt{2\gamma_{k}I} = R_{\gamma_{k}}(\cdot |X_{k-1})

Under a Foster-Lyapunov condition, R_{\gamma_{k}} is a Markov kernel that admits a unique stationary distribution \pi_{\gamma_{k}} that is close to the desired \pi in total variation distance and gets closer when \gamma_{k} decreases.

Now in the non-asymptotic case with fixed \gamma = \gamma_k, the authors provide bounds that explicitly depend on dimensionality of the support of the target, the number of samples drawn and the chosen step size \gamma . Unfortunately, these bounds contain some unknowns as well, such as the Lipschitz constant L of the gradient of the logpdf \log \pi(\cdot) and some suprema that I am unsure how to get explicitly.

Durmus and Moulines particularly take a look at scaling with dimension under increasingly strong conditions on \pi, getting exponential (in dimension) constants for the convergence when \pi is superexponential outside a ball. Better convergence can be achieved when assuming \pi to be log-concave or strongly log-concave. This is not surprising, nevertheless the theoretical importance of the results is clear from the fact that together with Arnak Dalalyan this is the first time that results are given for the ULA after the Roberts & Tweedie papers from 1996.

As a practitioner, I would have wished for very explicit guidance in picking  \gamma or the series \{\gamma_k\}_{k=1}^\infty. But hopefully with Alains paper as a foundation that can be the next step. As a non-mathematician, I had some problems in following the paper and at some point I completely lost it. This surely is in part due to the quite involved content. However, one might still manage to give intuition even in this case, as Sam Livingstones recent paper on HMC shows. I hope Alain goes over it again with readability and presentation in mind so that it will get the attention it deserves. Yet another task for something that already took a lot of work…

(photo: the Lipschitz Grill diner in Berlin – I don’t
know about their food, but the name is remarkable)

Importance Sampling Session at MCQMC 2016

On behalf of the mcqmc 2016 organizing committee I am pleased to accept your proposal.
-Art Owen
I got this nice message from Art yesterday night. My proposal for a session on Advances in Importance Sampling at MCQMC 2016 got accepted. Which is great, as I think the session is made up of strong papers (obviously). This session will almost surely be moderated by Nicolas Chopin.

MCQMC session on Advances in Importance Sampling

The sample size required in Importance Sampling

S. Chatterjee, P. Diaconis

The goal of importance sampling is to estimate the expected value of a given function with respect to a probability measure ν using a random sample of size n drawn from a different probability measure μ. If the two measures μ and ν are nearly singular with respect to each other, which is often the case in practice, the sample size required for accurate estimation is large. In this article it is shown that in a fairly general setting, a sample of size approximately exp(D(ν||μ)) is necessary and sufficient for accurate estimation by importance sampling, where D(ν||μ) is the Kullback–Leibler divergence of μ from ν. In particular, the required sample size exhibits a kind of cut-off in the logarithmic scale. The theory is applied to obtain a fairly general formula for the sample size required in importance sampling for exponential families (Gibbs measures). We also show that the standard variance-based diagnostic for convergence of importance sampling is fundamentally problematic. An alternative diagnostic that provably works in certain situations is suggested.

Generalized Multiple Importance Sampling

V. Elvira, L. Martino, D. Luengo, M. Bugallo

Importance Sampling methods are broadly used to approximate posterior distributions or some of their moments. In its standard approach, samples are drawn from a single proposal distribution and weighted properly. However, since the performance depends on the mismatch between the targeted and the proposal distributions, several proposal densities are often employed for the generation of samples. Under this Multiple Importance Sampling (MIS) scenario, many works have addressed the selection or adaptation of the proposal distributions, interpreting the sampling and the weighting steps in different ways. In this paper, we establish a general framework for sampling and weighing procedures when more than one proposal are available. The most relevant MIS schemes in the literature are encompassed within the new framework, and, moreover novel valid schemes appear naturally. All the MIS schemes are compared and ranked in terms of the variance of the associated estimators. Finally, we provide illustrative examples which reveal that, even with a good choice of the proposal densities, a careful interpretation of the sampling and weighting procedures can make a significant difference in the performance of the method.

Continuous-Time Importance Sampling

K. Łatuszyński, G. Roberts, G. Sermaidis, P. Fearnhead

We will introduce a new framework for sequential Monte Carlo, based on evolving a set of weighted particles in continuous time. This framework can lead to novel versions of existing algorithms, such as Annealed Importance Sampling and the Exact Algorithm for diffusions, and can be used as an alternative to MALA for sampling from a target distribution of interest. These methods are amenable to the use of sub-sampling, which can greatly increase their computational efficiency for big-data applications; and can enable unbiased sampling from a much wider-range of target distributions than existing approaches.

Quasi-Monte Carlo Methods in Finance

This second reference Mathieu Gerber gave me in the quest for educating myself about QMC, is paper by Pierre L’Ecuyer from the Winter Simulation Conference in 2004. It was much clearer as a tutorial (for me) as compared to the Art Owen paper. Maybe because it didn’t contain so much ANOVA. Or maybe because I was more used to ANOVA from Arts paper.

This paper specifically and quite transparently treats different constructions for low discrepancy point sets, in particular digital nets and their special cases. On the other hand, randomization procedures are discussed, which sometimes seem to be very specialized to the sequence used. One seemingly general transform after randomization called the baker transformation results in surprisingly high variance reduction of order O(n^{-4+\epsilon}). The transformation being to replace the uniform coordinate u \in [0,1) by 2u for u\leq 0.5 and 2(1-u) else.

In the examples L’Ecuyer mentions that using an Eigenzerlegung of covariance matrices (i.e. PCA) results in much higher variance reductions as compared to using Cholesky factors. Which he attributes to dimension reduction – a naming I find odd, as the complete information is retained (as opposed to, e.g. tossing the components with lowest Eigenvalue). My intuition is that maybe the strong empirical gains with PCA might rather be attributed to the fact that Eigenvectors are orthogonal, making this decomposition as close as possible to QMCs beloved unit hypercube.

Monte Carlo Extension of Quasi-Monte Carlo

405px-Subrandom_2D
Low discrepancy (top) vs. randomly uniform points (bottom)

Mathieu Gerber gave me this 1998 paper by Art Owen as one of several references for good introductions to QMC in the course of evaluating an idea we had at MCMSki.

The paper  surveys re-randomization of low discrepancy/QMC point sets mostly as a way of getting an estimate of the variance of the integral estimate. Or that is what the paper makes states most prominently – another reason for doing randomized QMC being that in some cases it further decreases variance as compared to plain QMC.

One of the things this paper stresses is that QMC will not help in truly high-dimensional integrands: say you have a d dimensional integrand which does not live on a lower dimensional manifold, and only use n \ll O(d^2). Then equidistributing these n points meaningfully in becomes impossible. Of course if the integrand does live on a lower dimensional manifold, one can use that fact to get convergence rates that correspond to the dimensionality of that manifold, which corresponds (informally) to what Art Owen calls effective dimension. Two definitions of effective dimension variants are given, both using the ANOVA decomposition. ANOVA only crossed my way earlier through my wifes psychology courses in stats, where it seemed to be a test mostly, so I’ll have delve more deeply into the decomposition method. It seems that basically, the value of the integrand f(x_1,\dots,x_d) is treated as a dependent variable while the x_1, \dots, x_d are the independent variables and ANOVA is used to get an idea of how independent variables interact in producing the dependent variable. In which case of course we would have to get some meaningful samples of points (x_1,\dots,x_d, f(x_1,\dots,x_d)) which currently seems to me like circling back to the beginning, since meaningful samples are what we want in the first place.

The idea for getting variance of the integral estimate from RQMC is the following: given a number m of deterministic QMC points, randomize them r times using independently drawn random variables. The integral estimates \hat I_1,\dots, \hat I_r are unbiased estimates of the true integral I with some variance \sigma_{\textrm{RQMC}}^2. Averaging over the $latex  \hat I_i$ we get another unbiased estimator $latex \hat I$ which has variance \sigma_{\textrm{RQMC}}^2/r. This RQMC variance can be estimated unbiasedly as \frac{1}{r(r-1)} \sum_{i=1}^r (\hat I_i - \hat I)^2. If m = 1 then r is the number of samples and this simply is the standard estimate of Monte Carlo variance.

The paper goes on to talk about using what it calls Latin Supercube Sampling for using RQMC in even higher dimensions, again using ANOVA decomposition and RQMC points in each group of interacting variables as determined by the ANOVA.

Overall, I know a bit more about QMC but am still more in the confusion than the knowledge state, which I hope Mathieus next reference will help with.