arXiv subjects

Arnaud Doucet

Publications and source records attributed to Arnaud Doucet.

At least 127 records · Page 7Linked to original sources

arXiv2020

Dual Space Preconditioning for Gradient Descent

The conditions of relative smoothness and relative strong convexity were recently introduced for the analysis of Bregman gradient methods for convex optimization. We introduce a generalized left-preconditioning method for gradient descent, and show that its convergence on an essentially smooth convex objective function can be guaranteed via an application of relative smoothness in the dual space. Our relative smoothness assumption is between the designed preconditioner and the convex conjugate of the objective, and it generalizes the typical Lipschitz gradient assumption. Under dual relative strong convexity, we obtain linear convergence with a generalized condition number that is invariant under horizontal translations, distinguishing it from Bregman gradient methods. Thus, in principle our method is capable of improving the conditioning of gradient descent on problems with non-Lipschitz gradient or non-strongly convex structure. We demonstrate our method on p-norm regression and exponential penalty function minimization.

math.OC↗

arXiv2020

Modular Meta-Learning with Shrinkage

Many real-world problems, including multi-speaker text-to-speech synthesis, can greatly benefit from the ability to meta-learn large models with only a few task-specific components. Updating only these task-specific modules then allows the model to be adapted to low-data tasks for as many steps as necessary without risking overfitting. Unfortunately, existing meta-learning methods either do not scale to long adaptation or else rely on handcrafted task-specific architectures. Here, we propose a meta-learning approach that obviates the need for this often sub-optimal hand-selection. In particular, we develop general techniques based on Bayesian shrinkage to automatically discover and learn both task-specific and general reusable modules. Empirically, we demonstrate that our method discovers a small set of meaningful task-specific modules and outperforms existing meta-learning approaches in domains like few-shot text-to-speech that have little task data and long adaptation horizons. We also show that existing meta-learning methods including MAML, iMAML, and Reptile emerge as special cases of our method.

cs.LG↗

arXiv2020

Noisy Adaptive Group Testing using Bayesian Sequential Experimental Design

When the infection prevalence of a disease is low, Dorfman showed 80 years ago that testing groups of people can prove more efficient than testing people individually. Our goal in this paper is to propose new group testing algorithms that can operate in a noisy setting (tests can be mistaken) to decide adaptively (looking at past results) which groups to test next, with the goal to converge to a good detection, as quickly, and with as few tests as possible. We cast this problem as a Bayesian sequential experimental design problem. Using the posterior distribution of infection status vectors for $n$ patients, given observed tests carried out so far, we seek to form groups that have a maximal utility. We consider utilities such as mutual information, but also quantities that have a more direct relevance to testing, such as the AUC of the ROC curve of the test. Practically, the posterior distributions on $\{0,1\}^n$ are approximated by sequential Monte Carlo (SMC) samplers and the utility maximized by a greedy optimizer. Our procedures show in simulations significant improvements over both adaptive and non-adaptive baselines, and are far more efficient than individual tests when disease prevalence is low. Additionally, we show empirically that loopy belief propagation (LBP), widely regarded as the SoTA decoder to decide whether an individual is infected or not given previous tests, can be unreliable and exhibit oscillatory behavior. Our SMC decoder is more reliable, and can improve the performance of other group testing algorithms.

stat.ME↗

arXiv2020

Unbiased Markov chain Monte Carlo for intractable target distributions

Performing numerical integration when the integrand itself cannot be evaluated point-wise is a challenging task that arises in statistical analysis, notably in Bayesian inference for models with intractable likelihood functions. Markov chain Monte Carlo (MCMC) algorithms have been proposed for this setting, such as the pseudo-marginal method for latent variable models and the exchange algorithm for a class of undirected graphical models. As with any MCMC algorithm, the resulting estimators are justified asymptotically in the limit of the number of iterations, but exhibit a bias for any fixed number of iterations due to the Markov chains starting outside of stationarity. This "burn-in" bias is known to complicate the use of parallel processors for MCMC computations. We show how to use coupling techniques to generate unbiased estimators in finite time, building on recent advances for generic MCMC algorithms. We establish the theoretical validity of some of these procedures by extending existing results to cover the case of polynomially ergodic Markov chains. The efficiency of the proposed estimators is compared with that of standard MCMC estimators, with theoretical arguments and numerical experiments including state space models and Ising models.

stat.CO↗

arXiv2020

Ensemble Rejection Sampling

We introduce Ensemble Rejection Sampling, a scheme for exact simulation from the posterior distribution of the latent states of a class of non-linear non-Gaussian state-space models. Ensemble Rejection Sampling relies on a proposal for the high-dimensional state sequence built using ensembles of state samples. Although this algorithm can be interpreted as a rejection sampling scheme acting on an extended space, we show under regularity conditions that the expected computational cost to obtain an exact sample increases cubically with the length of the state sequence instead of exponentially for standard rejection sampling. We demonstrate this methodology by sampling exactly state sequences according to the posterior distribution of a stochastic volatility model and a non-linear autoregressive process. We also present an application to rare event simulation.

stat.CO↗

arXiv2020

Gibbs flow for approximate transport with applications to Bayesian computation

Let $π_{0}$ and $π_{1}$ be two distributions on the Borel space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$. Any measurable function $T:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such that $Y=T(X)\simπ_{1}$ if $X\simπ_{0}$ is called a transport map from $π_{0}$ to $π_{1}$. For any $π_{0}$ and $π_{1}$, if one could obtain an analytical expression for a transport map from $π_{0}$ to $π_{1}$, then this could be straightforwardly applied to sample from any distribution. One would map draws from an easy-to-sample distribution $π_{0}$ to the target distribution $π_{1}$ using this transport map. Although it is usually impossible to obtain an explicit transport map for complex target distributions, we show here how to build a tractable approximation of a novel transport map. This is achieved by moving samples from $π_{0}$ using an ordinary differential equation with a velocity field that depends on the full conditional distributions of the target. Even when this ordinary differential equation is time-discretized and the full conditional distributions are numerically approximated, the resulting distribution of mapped samples can be efficiently evaluated and used as a proposal within sequential Monte Carlo samplers. We demonstrate significant gains over state-of-the-art sequential Monte Carlo samplers at a fixed computational complexity on a variety of applications.

stat.CO↗

arXiv2019

Schrödinger Bridge Samplers

Consider a reference Markov process with initial distribution $π_{0}$ and transition kernels $\{M_{t}\}_{t\in[1:T]}$, for some $T\in\mathbb{N}$. Assume that you are given distribution $π_{T}$, which is not equal to the marginal distribution of the reference process at time $T$. In this scenario, Schrödinger addressed the problem of identifying the Markov process with initial distribution $π_{0}$ and terminal distribution equal to $π_{T}$ which is the closest to the reference process in terms of Kullback--Leibler divergence. This special case of the so-called Schrödinger bridge problem can be solved using iterative proportional fitting, also known as the Sinkhorn algorithm. We leverage these ideas to develop novel Monte Carlo schemes, termed Schrödinger bridge samplers, to approximate a target distribution $π$ on $\mathbb{R}^{d}$ and to estimate its normalizing constant. This is achieved by iteratively modifying the transition kernels of the reference Markov chain to obtain a process whose marginal distribution at time $T$ becomes closer to $π_T = π$, via regression-based approximations of the corresponding iterative proportional fitting recursion. We report preliminary experiments and make connections with other problems arising in the optimal transport, optimal control and physics literatures.

stat.CO↗

arXiv2019

Stability of Optimal Filter Higher-Order Derivatives

In many scenarios, a state-space model depends on a parameter which needs to be inferred from data. Using stochastic gradient search and the optimal filter (first-order) derivative, the parameter can be estimated online. To analyze the asymptotic behavior of online methods for parameter estimation in non-linear state-space models, it is necessary to establish results on the existence and stability of the optimal filter higher-order derivatives. The existence and stability properties of these derivatives are studied here. We show that the optimal filter higher-order derivatives exist and forget initial conditions exponentially fast. We also show that the optimal filter higher-order derivatives are geometrically ergodic. The obtained results hold under (relatively) mild conditions and apply to state-space models met in practice.

math.PR↗

arXiv2019

Large Sample Asymptotics of the Pseudo-Marginal Method

The pseudo-marginal algorithm is a variant of the Metropolis--Hastings algorithm which samples asymptotically from a probability distribution when it is only possible to estimate unbiasedly an unnormalized version of its density. Practically, one has to trade-off the computational resources used to obtain this estimator against the asymptotic variances of the ergodic averages obtained by the pseudo-marginal algorithm. Recent works optimizing this trade-off rely on some strong assumptions which can cast doubts over their practical relevance. In particular, they all assume that the distribution of the difference between the log-density and its estimate is independent of the parameter value at which it is evaluated. Under regularity conditions we show here that, as the number of data points tends to infinity, a space-rescaled version of the pseudo-marginal chain converges weakly towards another pseudo-marginal chain for which this assumption indeed holds. A study of this limiting chain allows us to provide parameter dimension-dependent guidelines on how to optimally scale a normal random walk proposal and the number of Monte Carlo samples for the pseudo-marginal method in the large-sample regime. This complements and validates currently available results.

stat.CO↗

arXiv2019

Controlled Sequential Monte Carlo

Sequential Monte Carlo methods, also known as particle methods, are a popular set of techniques for approximating high-dimensional probability distributions and their normalizing constants. These methods have found numerous applications in statistics and related fields; e.g. for inference in non-linear non-Gaussian state space models, and in complex static models. Like many Monte Carlo sampling schemes, they rely on proposal distributions which crucially impact their performance. We introduce here a class of controlled sequential Monte Carlo algorithms, where the proposal distributions are determined by approximating the solution to an associated optimal control problem using an iterative scheme. This method builds upon a number of existing algorithms in econometrics, physics, and statistics for inference in state space models, and generalizes these methods so as to accommodate complex static models. We provide a theoretical analysis concerning the fluctuation and stability of this methodology that also provides insight into the properties of related algorithms. We demonstrate significant gains over state-of-the-art methods at a fixed computational complexity on a variety of applications.

stat.CO↗

arXiv2019

Augmented Neural ODEs

We show that Neural Ordinary Differential Equations (ODEs) learn representations that preserve the topology of the input space and prove that this implies the existence of functions Neural ODEs cannot represent. To address these limitations, we introduce Augmented Neural ODEs which, in addition to being more expressive models, are empirically more stable, generalize better and have a lower computational cost than Neural ODEs.

stat.ML↗

arXiv2019

Pseudo-Marginal Hamiltonian Monte Carlo

Bayesian inference in the presence of an intractable likelihood function is computationally challenging. When following a Markov chain Monte Carlo (MCMC) approach to approximate the posterior distribution in this context, one typically either uses MCMC schemes which target the joint posterior of the parameters and some auxiliary latent variables, or pseudo-marginal Metropolis--Hastings (MH) schemes. The latter mimic a MH algorithm targeting the marginal posterior of the parameters by approximating unbiasedly the intractable likelihood. However, in scenarios where the parameters and auxiliary variables are strongly correlated under the posterior and/or this posterior is multimodal, Gibbs sampling or Hamiltonian Monte Carlo (HMC) will perform poorly and the pseudo-marginal MH algorithm, as any other MH scheme, will be inefficient for high dimensional parameters. We propose here an original MCMC algorithm, termed pseudo-marginal HMC, which combines the advantages of both HMC and pseudo-marginal schemes. Specifically, the pseudo-marginal HMC method is controlled by a precision parameter N, controlling the approximation of the likelihood and, for any N, it samples the marginal posterior of the parameters. Additionally, as N tends to infinity, its sample trajectories and acceptance probability converge to those of an ideal, but intractable, HMC algorithm which would have access to the marginal posterior of parameters and its gradient. We demonstrate through experiments that pseudo-marginal HMC can outperform significantly both standard HMC and pseudo-marginal MH schemes.

stat.ME↗

arXiv2019

Analyticity of Entropy Rates of Continuous-State Hidden Markov Models

The analyticity of the entropy and relative entropy rates of continuous-state hidden Markov models is studied here. Using the analytic continuation principle and the stability properties of the optimal filter, the analyticity of these rates is shown for analytically parameterized models. The obtained results hold under relatively mild conditions and cover several classes of hidden Markov models met in practice. These results are relevant for several (theoretically and practically) important problems arising in statistical inference, system identification and information theory.

cs.IT↗

arXiv2019

Scalable Metropolis-Hastings for Exact Bayesian Inference with Large Datasets

Bayesian inference via standard Markov Chain Monte Carlo (MCMC) methods is too computationally intensive to handle large datasets, since the cost per step usually scales like $Θ(n)$ in the number of data points $n$. We propose the Scalable Metropolis-Hastings (SMH) kernel that exploits Gaussian concentration of the posterior to require processing on average only $O(1)$ or even $O(1/\sqrt{n})$ data points per step. This scheme is based on a combination of factorized acceptance probabilities, procedures for fast simulation of Bernoulli processes, and control variate ideas. Contrary to many MCMC subsampling schemes such as fixed step-size Stochastic Gradient Langevin Dynamics, our approach is exact insofar as the invariant distribution is the true posterior and not an approximation to it. We characterise the performance of our algorithm theoretically, and give realistic and verifiable conditions under which it is geometrically ergodic. This theory is borne out by empirical results that demonstrate overall performance benefits over standard Metropolis-Hastings and various subsampling algorithms.

stat.ML↗

arXiv2019

On the Impact of the Activation Function on Deep Neural Networks Training

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 Samuel 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. While the work by Samuel et al (2017) discuss trainability issues, we focus here on training acceleration and overall performance. We give a comprehensive theoretical analysis of the Edge of Chaos and show that we can indeed tune the initialization parameters and the activation function in order to accelerate the training and improve the performance.

stat.ML↗

arXiv2019

Replica Conditional Sequential Monte Carlo

We propose a Markov chain Monte Carlo (MCMC) scheme to perform state inference in non-linear non-Gaussian state-space models. Current state-of-the-art methods to address this problem rely on particle MCMC techniques and its variants, such as the iterated conditional Sequential Monte Carlo (cSMC) scheme, which uses a Sequential Monte Carlo (SMC) type proposal within MCMC. A deficiency of standard SMC proposals is that they only use observations up to time $t$ to propose states at time $t$ when an entire observation sequence is available. More sophisticated SMC based on lookahead techniques could be used but they can be difficult to put in practice. We propose here replica cSMC where we build SMC proposals for one replica using information from the entire observation sequence by conditioning on the states of the other replicas. This approach is easily parallelizable and we demonstrate its excellent empirical performance when compared to the standard iterated cSMC scheme at fixed computational complexity.

stat.CO↗

arXiv2019

Bernoulli Race Particle Filters

When the weights in a particle filter are not available analytically, standard resampling methods cannot be employed. To circumvent this problem state-of-the-art algorithms replace the true weights with non-negative unbiased estimates. This algorithm is still valid but at the cost of higher variance of the resulting filtering estimates in comparison to a particle filter using the true weights. We propose here a novel algorithm that allows for resampling according to the true intractable weights when only an unbiased estimator of the weights is available. We demonstrate our algorithm on several examples.

stat.CO↗

arXiv2019

Unbiased Smoothing using Particle Independent Metropolis-Hastings

We consider the approximation of expectations with respect to the distribution of a latent Markov process given noisy measurements. This is known as the smoothing problem and is often approached with particle and Markov chain Monte Carlo (MCMC) methods. These methods provide consistent but biased estimators when run for a finite time. We propose a simple way of coupling two MCMC chains built using Particle Independent Metropolis-Hastings (PIMH) to produce unbiased smoothing estimators. Unbiased estimators are appealing in the context of parallel computing, and facilitate the construction of confidence intervals. The proposed scheme only requires access to off-the-shelf Particle Filters (PF) and is thus easier to implement than recently proposed unbiased smoothers. The approach is demonstrated on a Lévy-driven stochastic volatility model and a stochastic kinetic model.

stat.CO↗