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Arnaud Doucet

Publications and source records attributed to Arnaud Doucet.

At least 145 records · Page 8Linked to original sources
arXiv2018

Hamiltonian Variational Auto-Encoder

Variational Auto-Encoders (VAEs) have become very popular techniques to perform inference and learning in latent variable models as they allow us to leverage the rich representational power of neural networks to obtain flexible approximations of the posterior of latent variables as well as tight evidence lower bounds (ELBOs). Combined with stochastic variational inference, this provides a methodology scaling to large datasets. However, for this methodology to be practically efficient, it is necessary to obtain low-variance unbiased estimators of the ELBO and its gradients with respect to the parameters of interest. While the use of Markov chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo (HMC) has been previously suggested to achieve this [23, 26], the proposed methods require specifying reverse kernels which have a large impact on performance. Additionally, the resulting unbiased estimator of the ELBO for most MCMC kernels is typically not amenable to the reparameterization trick. We show here how to optimally select reverse kernels in this setting and, by building upon Hamiltonian Importance Sampling (HIS) [17], we obtain a scheme that provides low-variance unbiased estimators of the ELBO and its gradients using the reparameterization trick. This allows us to develop a Hamiltonian Variational Auto-Encoder (HVAE). This method can be reinterpreted as a target-informed normalizing flow [20] which, within our context, only requires a few evaluations of the gradient of the sampled likelihood and trivial Jacobian calculations at each iteration.

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arXiv2018

On the Selection of Initialization and Activation Function for Deep Neural Networks

The weight initialization and the activation function of deep neural networks have a crucial impact on the performance of the training procedure. An inappropriate selection can lead to the loss of information of the input during forward propagation and the exponential vanishing/exploding of gradients during back-propagation. Understanding the theoretical properties of untrained random networks is key to identifying which deep networks may be trained successfully as recently demonstrated by Schoenholz et al. (2017) who showed that for deep feedforward neural networks only a specific choice of hyperparameters known as the `edge of chaos' can lead to good performance. We complete this analysis by providing quantitative results showing that, for a class of ReLU-like activation functions, the information propagates indeed deeper for an initialization at the edge of chaos. By further extending this analysis, we identify a class of activation functions that improve the information propagation over ReLU-like functions. This class includes the Swish activation, $ϕ_{swish}(x) = x \cdot \text{sigmoid}(x)$, used in Hendrycks & Gimpel (2016), Elfwing et al. (2017) and Ramachandran et al. (2017). This provides a theoretical grounding for the excellent empirical performance of $ϕ_{swish}$ observed in these contributions. We complement those previous results by illustrating the benefit of using a random initialization on the edge of chaos in this context.

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arXiv2018

Hamiltonian Descent Methods

We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger class includes functions whose second derivatives may be singular or unbounded at their minima. Our methods are discretizations of conformal Hamiltonian dynamics, which generalize the classical momentum method to model the motion of a particle with non-standard kinetic energy exposed to a dissipative force and the gradient field of the function of interest. They are first-order in the sense that they require only gradient computation. Yet, crucially the kinetic gradient map can be designed to incorporate information about the convex conjugate in a fashion that allows for linear convergence on convex functions that may be non-smooth or non-strongly convex. We study in detail one implicit and two explicit methods. For one explicit method, we provide conditions under which it converges to stationary points of non-convex functions. For all, we provide conditions on the convex function and kinetic energy pair that guarantee linear convergence, and show that these conditions can be satisfied by functions with power growth. In sum, these methods expand the class of convex functions on which linear convergence is possible with first-order computation.

math.OC↗
arXiv2018

Scalable Monte Carlo inference for state-space models

We present an original simulation-based method to estimate likelihood ratios efficiently for general state-space models. Our method relies on a novel use of the conditional Sequential Monte Carlo (cSMC) algorithm introduced in \citet{Andrieu_et_al_2010} and presents several practical advantages over standard approaches. The ratio is estimated using a unique source of randomness instead of estimating separately the two likelihood terms involved. Beyond the benefits in terms of variance reduction one may expect in general from this type of approach, an important point here is that the variance of this estimator decreases as the distance between the likelihood parameters decreases. We show how this can be exploited in the context of Monte Carlo Markov chain (MCMC) algorithms, leading to the development of a new class of exact-approximate MCMC methods to perform Bayesian static parameter inference in state-space models. We show through simulations that, in contrast to the Particle Marginal Metropolis-Hastings (PMMH) algorithm of Andrieu_et_al_2010, the computational effort required by this novel MCMC scheme scales very favourably for large data sets.

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arXiv2018

Limit theorems for sequential MCMC methods

Sequential Monte Carlo (SMC) methods, also known as particle filters, constitute a class of algorithms used to approximate expectations with respect to a sequence of probability distributions as well as the normalising constants of those distributions. Sequential MCMC methods are an alternative class of techniques addressing similar problems in which particles are sampled according to an MCMC kernel rather than conditionally independently at each time step. These methods were introduced over twenty years ago by Berzuini et al. (1997). Recently, there has been a renewed interest in such algorithms as they demonstrate an empirical performance superior to that of SMC methods in some applications. We establish a strong law of large numbers and a central limit theorem for sequential MCMC methods and provide conditions under which errors can be controlled uniformly in time. In the context of state-space models, we provide conditions under which sequential MCMC methods can indeed outperform standard SMC methods in terms of asymptotic variance of the corresponding Monte Carlo estimators.

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arXiv2018

Piecewise-Deterministic Markov Chain Monte Carlo

A novel class of non-reversible Markov chain Monte Carlo schemes relying on continuous-time piecewise-deterministic Markov Processes has recently emerged. In these algorithms, the state of the Markov process evolves according to a deterministic dynamics which is modified using a Markov transition kernel at random event times. These methods enjoy remarkable features including the ability to update only a subset of the state components while other components implicitly keep evolving and the ability to use an unbiased estimate of the gradient of the log-target while preserving the target as invariant distribution. However, they also suffer from important limitations. The deterministic dynamics used so far do not exploit the structure of the target. Moreover, exact simulation of the event times is feasible for an important yet restricted class of problems and, even when it is, it is application specific. This limits the applicability of these techniques and prevents the development of a generic software implementation of them. We introduce novel MCMC methods addressing these shortcomings. In particular, we introduce novel continuous-time algorithms relying on exact Hamiltonian flows and novel non-reversible discrete-time algorithms which can exploit complex dynamics such as approximate Hamiltonian dynamics arising from symplectic integrators while preserving the attractive features of continuous-time algorithms. We demonstrate the performance of these schemes on a variety of applications.

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arXiv2018

On the utility of Metropolis-Hastings with asymmetric acceptance ratio

The Metropolis-Hastings algorithm allows one to sample asymptotically from any probability distribution $π$. There has been recently much work devoted to the development of variants of the MH update which can handle scenarios where such an evaluation is impossible, and yet are guaranteed to sample from $π$ asymptotically. The most popular approach to have emerged is arguably the pseudo-marginal MH algorithm which substitutes an unbiased estimate of an unnormalised version of $π$ for $π$. Alternative pseudo-marginal algorithms relying instead on unbiased estimates of the MH acceptance ratio have also been proposed. These algorithms can have better properties than standard PM algorithms. Convergence properties of both classes of algorithms are known to depend on the variability of the estimators involved and reduced variability is guaranteed to decrease the asymptotic variance of ergodic averages and will shorten the burn-in period, or convergence to equilibrium, in most scenarios of interest. A simple approach to reduce variability, amenable to parallel computations, consists of averaging independent estimators. However, while averaging estimators of $π$ in a pseudo-marginal algorithm retains the guarantee of sampling from $π$ asymptotically, naive averaging of acceptance ratio estimates breaks detailed balance, leading to incorrect results. We propose an original methodology which allows for a correct implementation of this idea. We establish theoretical properties which parallel those available for standard PM algorithms and discussed above. We demonstrate the interest of the approach on various inference problems. In particular we show that convergence to equilibrium can be significantly shortened, therefore offering the possibility to reduce a user's waiting time in a generic fashion when a parallel computing architecture is available.

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arXiv2017

Filtering Variational Objectives

When used as a surrogate objective for maximum likelihood estimation in latent variable models, the evidence lower bound (ELBO) produces state-of-the-art results. Inspired by this, we consider the extension of the ELBO to a family of lower bounds defined by a particle filter's estimator of the marginal likelihood, the filtering variational objectives (FIVOs). FIVOs take the same arguments as the ELBO, but can exploit a model's sequential structure to form tighter bounds. We present results that relate the tightness of FIVO's bound to the variance of the particle filter's estimator by considering the generic case of bounds defined as log-transformed likelihood estimators. Experimentally, we show that training with FIVO results in substantial improvements over training the same model architecture with the ELBO on sequential data.

cs.LG↗
arXiv2017

Asymptotic Bias of Stochastic Gradient Search

The asymptotic behavior of the stochastic gradient algorithm with a biased gradient estimator is analyzed. Relying on arguments based on the dynamic system theory (chain-recurrence) and the differential geometry (Yomdin theorem and Lojasiewicz inequality), tight bounds on the asymptotic bias of the iterates generated by such an algorithm are derived. The obtained results hold under mild conditions and cover a broad class of high-dimensional nonlinear algorithms. Using these results, the asymptotic properties of the policy-gradient (reinforcement) learning and adaptive population Monte Carlo sampling are studied. Relying on the same results, the asymptotic behavior of the recursive maximum split-likelihood estimation in hidden Markov models is analyzed, too.

math.ST↗
arXiv2017

Exponential Ergodicity of the Bouncy Particle Sampler

Non-reversible Markov chain Monte Carlo schemes based on piecewise deterministic Markov processes have been recently introduced in applied probability, automatic control, physics and statistics. Although these algorithms demonstrate experimentally good performance and are accordingly increasingly used in a wide range of applications, geometric ergodicity results for such schemes have only been established so far under very restrictive assumptions. We give here verifiable conditions on the target distribution under which the Bouncy Particle Sampler algorithm introduced in \cite{P_dW_12} is geometrically ergodic. This holds whenever the target satisfies a curvature condition and has tails decaying at least as fast as an exponential and at most as fast as a Gaussian distribution. This allows us to provide a central limit theorem for the associated ergodic averages. When the target has tails thinner than a Gaussian distribution, we propose an original modification of this scheme that is geometrically ergodic. For thick-tailed target distributions, such as $t$-distributions, we extend the idea pioneered in \cite{J_G_12} in a random walk Metropolis context. We apply a change of variable to obtain a transformed target satisfying the tail conditions for geometric ergodicity. By sampling the transformed target using the Bouncy Particle Sampler and mapping back the Markov process to the original parameterization, we obtain a geometrically ergodic algorithm.

stat.CO↗
arXiv2017

The Correlated Pseudo-Marginal Method

The pseudo-marginal algorithm is a popular variant of the Metropolis--Hastings scheme which allows us to sample asymptotically from a target probability density $π$, when we are only able to estimate an unnormalized version of $π$ pointwise unbiasedly. It has found numerous applications in Bayesian statistics as there are many scenarios where the likelihood function is intractable but can be estimated unbiasedly using Monte Carlo samples. Using many samples will typically result in averages computed under this chain with lower asymptotic variances than the corresponding averages that use fewer samples. For a fixed computing time, it has been shown in several recent contributions that an efficient implementation of the pseudo-marginal method requires the variance of the log-likelihood ratio estimator appearing in the acceptance probability of the algorithm to be of order 1, which in turn usually requires scaling the number $N$ of Monte Carlo samples linearly with the number $T$ of data points. We propose a modification of the pseudo-marginal algorithm, termed the correlated pseudo-marginal algorithm, which is based on a novel log-likelihood ratio estimator computed using the difference of two positively correlated log-likelihood estimators. We show that the parameters of this scheme can be selected such that the variance of this estimator is order $1$ as $N,T\rightarrow\infty$ whenever $N/T\rightarrow 0$. By combining these results with the Bernstein-von Mises theorem, we provide an analysis of the performance of the correlated pseudo-marginal algorithm in the large $T$ regime. In our numerical examples, the efficiency of computations is increased relative to the standard pseudo-marginal algorithm by more than 20 fold for values of $T$ of a few hundreds to more than 100 fold for values of $T$ of around 10,000-20,000.

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arXiv2017

Particle Gibbs Split-Merge Sampling for Bayesian Inference in Mixture Models

This paper presents a new Markov chain Monte Carlo method to sample from the posterior distribution of conjugate mixture models. This algorithm relies on a flexible split-merge procedure built using the particle Gibbs sampler. Contrary to available split-merge procedures, the resulting so-called Particle Gibbs Split-Merge sampler does not require the computation of a complex acceptance ratio, is simple to implement using existing sequential Monte Carlo libraries and can be parallelized. We investigate its performance experimentally on synthetic problems as well as on geolocation and cancer genomics data. In all these examples, the particle Gibbs split-merge sampler outperforms state-of-the-art split-merge methods by up to an order of magnitude for a fixed computational complexity.

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arXiv2017

Interacting Particle Markov Chain Monte Carlo

We introduce interacting particle Markov chain Monte Carlo (iPMCMC), a PMCMC method based on an interacting pool of standard and conditional sequential Monte Carlo samplers. Like related methods, iPMCMC is a Markov chain Monte Carlo sampler on an extended space. We present empirical results that show significant improvements in mixing rates relative to both non-interacting PMCMC samplers, and a single PMCMC sampler with an equivalent memory and computational budget. An additional advantage of the iPMCMC method is that it is suitable for distributed and multi-core architectures.

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arXiv2017

Particle Value Functions

The policy gradients of the expected return objective can react slowly to rare rewards. Yet, in some cases agents may wish to emphasize the low or high returns regardless of their probability. Borrowing from the economics and control literature, we review the risk-sensitive value function that arises from an exponential utility and illustrate its effects on an example. This risk-sensitive value function is not always applicable to reinforcement learning problems, so we introduce the particle value function defined by a particle filter over the distributions of an agent's experience, which bounds the risk-sensitive one. We illustrate the benefit of the policy gradients of this objective in Cliffworld.

cs.LG↗
arXiv2017

The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method

Markov chain Monte Carlo methods have become standard tools in statistics to sample from complex probability measures. Many available techniques rely on discrete-time reversible Markov chains whose transition kernels build up over the Metropolis-Hastings algorithm. We explore and propose several original extensions of an alternative approach introduced recently in Peters and de With (2012) where the target distribution of interest is explored using a continuous-time Markov process. In the Metropolis-Hastings algorithm, a trial move to a region of lower target density, equivalently "higher energy", than the current state can be rejected with positive probability. In this alternative approach, a particle moves along straight lines continuously around the space and, when facing a high energy barrier, it is not rejected but its path is modified by bouncing against this barrier. The resulting non-reversible Markov process provides a rejection-free MCMC sampling scheme. We propose several original techniques to simulate this continuous-time process exactly in a wide range of scenarios of interest to statisticians. When the target distribution factorizes as a product of factors involving only subsets of variables, such as the posterior distribution associated to a probabilistic graphical model, it is possible to modify the original algorithm to exploit this structure and update in parallel variables within each clique. We present several extensions by proposing methods to sample mixed discrete-continuous distributions and distributions restricted to a connected smooth domain. We also show that it is possible to move the particle using a general flow instead of straight lines. We demonstrate the efficiency of this methodology through simulations on a variety of applications and show that it can outperform Hybrid Monte Carlo schemes in interesting scenarios.

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arXiv2016

On embedded hidden Markov models and particle Markov chain Monte Carlo methods

The embedded hidden Markov model (EHMM) sampling method is a Markov chain Monte Carlo (MCMC) technique for state inference in non-linear non-Gaussian state-space models which was proposed in Neal (2003); Neal et al. (2004) and extended in Shestopaloff and Neal (2016). An extension to Bayesian parameter inference was presented in Shestopaloff and Neal (2013). An alternative class of MCMC schemes addressing similar inference problems is provided by particle MCMC (PMCMC) methods (Andrieu et al. 2009; 2010). All these methods rely on the introduction of artificial extended target distributions for multiple state sequences which, by construction, are such that one randomly indexed sequence is distributed according to the posterior of interest. By adapting the Metropolis-Hastings algorithms developed in the framework of PMCMC methods to the EHMM framework, we obtain novel particle filter (PF)-type algorithms for state inference and novel MCMC schemes for parameter and state inference. In addition, we show that most of these algorithms can be viewed as particular cases of a general PF and PMCMC framework. We compare the empirical performance of the various algorithms on low- to high-dimensional state-space models. We demonstrate that a properly tuned conditional PF with "local" MCMC moves proposed in Shestopaloff and Neal (2016) can outperform the standard conditional PF significantly when applied to high-dimensional state-space models while the novel PF-type algorithm could prove to be an interesting alternative to standard PFs for likelihood estimation in some lower-dimensional scenarios.

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arXiv2016

Bayesian nonparametric image segmentation using a generalized Swendsen-Wang algorithm

Unsupervised image segmentation aims at clustering the set of pixels of an image into spatially homogeneous regions. We introduce here a class of Bayesian nonparametric models to address this problem. These models are based on a combination of a Potts-like spatial smoothness component and a prior on partitions which is used to control both the number and size of clusters. This class of models is flexible enough to include the standard Potts model and the more recent Potts-Dirichlet Process model \cite{Orbanz2008}. More importantly, any prior on partitions can be introduced to control the global clustering structure so that it is possible to penalize small or large clusters if necessary. Bayesian computation is carried out using an original generalized Swendsen-Wang algorithm. Experiments demonstrate that our method is competitive in terms of RAND\ index compared to popular image segmentation methods, such as mean-shift, and recent alternative Bayesian nonparametric models.

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