arXiv subjects

Judith Rousseau

Publications and source records attributed to Judith Rousseau.

At least 37 records · Page 2Linked to original sources

arXiv2023

Fast Bayesian Coresets via Subsampling and Quasi-Newton Refinement

Bayesian coresets approximate a posterior distribution by building a small weighted subset of the data points. Any inference procedure that is too computationally expensive to be run on the full posterior can instead be run inexpensively on the coreset, with results that approximate those on the full data. However, current approaches are limited by either a significant run-time or the need for the user to specify a low-cost approximation to the full posterior. We propose a Bayesian coreset construction algorithm that first selects a uniformly random subset of data, and then optimizes the weights using a novel quasi-Newton method. Our algorithm is a simple to implement, black-box method, that does not require the user to specify a low-cost posterior approximation. It is the first to come with a general high-probability bound on the KL divergence of the output coreset posterior. Experiments demonstrate that our method provides significant improvements in coreset quality against alternatives with comparable construction times, with far less storage cost and user input required.

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arXiv2022

On sparsity, power-law and clustering properties of graphex processes

This paper investigates properties of the class of graphs based on exchangeable point processes. We provide asymptotic expressions for the number of edges, number of nodes and degree distributions, identifying four regimes: (i) a dense regime, (ii) a sparse almost dense regime, (iii) a sparse regime with power-law behaviour, and (iv) an almost extremely sparse regime. We show that under mild assumptions, both the global and local clustering coefficients converge to constants which may or may not be the same. We also derive a central limit theorem for the number of nodes. Finally, we propose a class of models within this framework where one can separately control the latent structure and the global sparsity/power-law properties of the graph.

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arXiv2022

A flexible, random histogram kernel for discrete-time Hawkes processes

Hawkes processes are a self-exciting stochastic process used to describe phenomena whereby past events increase the probability of the occurrence of future events. This work presents a flexible approach for modelling a variant of these, namely discrete-time Hawkes processes. Most standard models of Hawkes processes rely on a parametric form for the function describing the influence of past events, referred to as the triggering kernel. This is likely to be insufficient to capture the true excitation pattern, particularly for complex data. By utilising trans-dimensional Markov chain Monte Carlo inference techniques, our proposed model for the triggering kernel can take the form of any step function, affording significantly more flexibility than a parametric form. We first demonstrate the utility of the proposed model through a comprehensive simulation study. This includes univariate scenarios, and multivariate scenarios whereby there are multiple interacting Hawkes processes. We then apply the proposed model to several case studies: the interaction between two countries during the early to middle stages of the COVID-19 pandemic, taking Italy and France as an example, and the interaction of terrorist activity between two countries in close spatial proximity, Indonesia and the Philippines, and then within three regions of the Philippines.

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arXiv2022

Exact Convergence Rates of the Neural Tangent Kernel in the Large Depth Limit

Recent work by Jacot et al. (2018) has shown that training a neural network using gradient descent in parameter space is related to kernel gradient descent in function space with respect to the Neural Tangent Kernel (NTK). Lee et al. (2019) built on this result by establishing that the output of a neural network trained using gradient descent can be approximated by a linear model when the network width is large. Indeed, under regularity conditions, the NTK converges to a time-independent kernel in the infinite-width limit. This regime is often called the NTK regime. In parallel, recent works on signal propagation (Poole et al., 2016; Schoenholz et al., 2017; Hayou et al., 2019a) studied the impact of the initialization and the activation function on signal propagation in deep neural networks. In this paper, we connect these two theories by quantifying the impact of the initialization and the activation function on the NTK when the network depth becomes large. In particular, we provide a comprehensive analysis of the convergence rates of the NTK regime to the infinite depth regime.

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arXiv2022

Asymptotic Analysis of Statistical Estimators related to MultiGraphex Processes under Misspecification

This article studies the asymptotic properties of Bayesian or frequentist estimators of a vector of parameters related to structural properties of sequences of graphs. The estimators studied originate from a particular class of graphex model introduced by Caron and Fox. The analysis is however performed here under very weak assumptions on the underlying data generating process, which may be different from the model of Caron and Fox or from a graphex model. In particular, we consider generic sparse graph models, with unbounded degree, whose degree distribution satisfies some assumptions. We show that one can relate the limit of the estimator of one of the parameters to the sparsity constant of the true graph generating process. When taking a Bayesian approach, we also show that the posterior distribution is asymptotically normal. We discuss situations where classical random graphs models such as configuration models, sparse graphon models, edge exchangeable models or graphon processes satisfy our assumptions.

math.ST↗

arXiv2022

Evidence estimation in finite and infinite mixture models and applications

Estimating the model evidence - or mariginal likelihood of the data - is a notoriously difficult task for finite and infinite mixture models and we reexamine here different Monte Carlo techniques advocated in the recent literature, as well as novel approaches based on Geyer (1994) reverse logistic regression technique, Chib (1995) algorithm, and Sequential Monte Carlo (SMC). Applications are numerous. In particular, testing for the number of components in a finite mixture model or against the fit of a finite mixture model for a given dataset has long been and still is an issue of much interest, albeit yet missing a fully satisfactory resolution. Using a Bayes factor to find the right number of components K in a finite mixture model is known to provide a consistent procedure. We furthermore establish the consistence of the Bayes factor when comparing a parametric family of finite mixtures against the nonparametric 'strongly identifiable' Dirichlet Process Mixture (DPM) model.

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arXiv2022

Bayesian Nonparametrics for Sparse Dynamic Networks

In this paper we propose a Bayesian nonparametric approach to modelling sparse time-varying networks. A positive parameter is associated to each node of a network, which models the sociability of that node. Sociabilities are assumed to evolve over time, and are modelled via a dynamic point process model. The model is able to capture long term evolution of the sociabilities. Moreover, it yields sparse graphs, where the number of edges grows subquadratically with the number of nodes. The evolution of the sociabilities is described by a tractable time-varying generalised gamma process. We provide some theoretical insights into the model and apply it to three datasets: a simulated network, a network of hyperlinks between communities on Reddit, and a network of co-occurences of words in Reuters news articles after the September 11th attacks.

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arXiv2021

Wasserstein convergence in Bayesian deconvolution models

We study the reknown deconvolution problem of recovering a distribution function from independent replicates (signal) additively contaminated with random errors (noise), whose distribution is known. We investigate whether a Bayesian nonparametric approach for modelling the latent distribution of the signal can yield inferences with asymptotic frequentist validity under the $L^1$-Wasserstein metric. When the error density is ordinary smooth, we develop two inversion inequalities relating either the $L^1$ or the $L^1$-Wasserstein distance between two mixture densities (of the observations) to the $L^1$-Wasserstein distance between the corresponding distributions of the signal. This smoothing inequality improves on those in the literature. We apply this general result to a Bayesian approach bayes on a Dirichlet process mixture of normal distributions as a prior on the mixing distribution (or distribution of the signal), with a Laplace or Linnik noise. In particular we construct an \textit{adaptive} approximation of the density of the observations by the convolution of a Laplace (or Linnik) with a well chosen mixture of normal densities and show that the posterior concentrates at the minimax rate up to a logarithmic factor. The same prior law is shown to also adapt to the Sobolev regularity level of the mixing density, thus leading to a new Bayesian estimation method, relative to the Wasserstein distance, for distributions with smooth densities.

math.ST↗

arXiv2021

Ideal Bayesian Spatial Adaptation

Many real-life applications involve estimation of curves that exhibit complicated shapes including jumps or varying-frequency oscillations. Practical methods have been devised that can adapt to a locally varying complexity of an unknown function (e.g. variable-knot splines, sparse wavelet reconstructions, kernel methods or trees/forests). However, the overwhelming majority of existing asymptotic minimaxity theory is predicated on homogeneous smoothness assumptions. Focusing on locally Holderian functions, we provide new locally adaptive posterior concentration rate results under the supremum loss for widely used Bayesian machine learning techniques in white noise and non-parametric regression. In particular, we show that popular spike-and-slab priors and Bayesian CART are uniformly locally adaptive. In addition, we propose a new class of repulsive partitioning priors which relate to variable knot splines and which are exact-rate adaptive. For uncertainty quantification, we construct locally adaptive confidence bands whose width depends on the local smoothness and which achieve uniform asymptotic coverage under local self-similarity. To illustrate that spatial adaptation is not at all automatic, we provide lower-bound results showing that popular hierarchical Gaussian process priors fall short of spatial adaptation.

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arXiv2021

Stable ResNet

Deep ResNet architectures have achieved state of the art performance on many tasks. While they solve the problem of gradient vanishing, they might suffer from gradient exploding as the depth becomes large (Yang et al. 2017). Moreover, recent results have shown that ResNet might lose expressivity as the depth goes to infinity (Yang et al. 2017, Hayou et al. 2019). To resolve these issues, we introduce a new class of ResNet architectures, called Stable ResNet, that have the property of stabilizing the gradient while ensuring expressivity in the infinite depth limit.

cs.LG↗

arXiv2019

Sparse Networks with Core-Periphery Structure

We propose a statistical model for graphs with a core-periphery structure. To do this we define a precise notion of what it means for a graph to have this structure, based on the sparsity properties of the subgraphs of core and periphery nodes. We present a class of sparse graphs with such properties, and provide methods to simulate from this class, and to perform posterior inference. We demonstrate that our model can detect core-periphery structure in simulated and real-world networks.

stat.ME↗

arXiv2019

Model Misspecification in ABC: Consequences and Diagnostics

We analyze the behavior of approximate Bayesian computation (ABC) when the model generating the simulated data differs from the actual data generating process; i.e., when the data simulator in ABC is misspecified. We demonstrate both theoretically and in simple, but practically relevant, examples that when the model is misspecified different versions of ABC can yield substantially different results. Our theoretical results demonstrate that even though the model is misspecified, under regularity conditions, the accept/reject ABC approach concentrates posterior mass on an appropriately defined pseudo-true parameter value. However, under model misspecification the ABC posterior does not yield credible sets with valid frequentist coverage and has non-standard asymptotic behavior. In addition, we examine the theoretical behavior of the popular local regression adjustment to ABC under model misspecification and demonstrate that this approach concentrates posterior mass on a completely different pseudo-true value than accept/reject ABC. Using our theoretical results, we suggest two approaches to diagnose model misspecification in ABC. All theoretical results and diagnostics are illustrated in a simple running example.

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arXiv2019

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

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arXiv2019

Asymptotic frequentist coverage properties of Bayesian credible sets for sieve priors

We investigate the frequentist coverage properties of Bayesian credible sets in a general, adaptive, nonparametric framework. It is well known that the construction of adaptive and honest confidence sets is not possible in general. To overcome this problem we introduce an extra assumption on the functional parameters, the so called "general polished tail" condition. We then show that under standard assumptions both the hierarchical and empirical Bayes methods results in honest confidence sets for sieve type of priors in general settings and we characterize their size. We apply the derived abstract results to various examples, including the nonparametric regression model, density estimation using exponential families of priors, density estimation using histogram priors and nonparametric classification model, for which we show that their size is near minimax adaptive with respect to the considered specific semi-metrics.

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arXiv2018

Testing hypotheses via a mixture estimation model

We consider a novel paradigm for Bayesian testing of hypotheses and Bayesian model comparison. Our alternative to the traditional construction of posterior probabilities that a given hypothesis is true or that the data originates from a specific model is to consider the models under comparison as components of a mixture model. We therefore replace the original testing problem with an estimation one that focus on the probability weight of a given model within a mixture model. We analyze the sensitivity on the resulting posterior distribution on the weights of various prior modeling on the weights. We stress that a major appeal in using this novel perspective is that generic improper priors are acceptable, while not putting convergence in jeopardy. Among other features, this allows for a resolution of the Lindley-Jeffreys paradox. When using a reference Beta B(a,a) prior on the mixture weights, we note that the sensitivity of the posterior estimations of the weights to the choice of a vanishes with the sample size increasing and avocate the default choice a=0.5, derived from Rousseau and Mengersen (2011). Another feature of this easily implemented alternative to the classical Bayesian solution is that the speeds of convergence of the posterior mean of the weight and of the corresponding posterior probability are quite similar.

stat.ME↗

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.

stat.ML↗

arXiv2018

Adaptive Supremum Norm Posterior Contraction: Wavelet Spike-and-Slab and Anisotropic Besov Spaces

Supremum norm loss is intuitively more meaningful to quantify function estimation error in statistics. In the context of multivariate nonparametric regression with unknown error, we propose a Bayesian procedure based on spike-and-slab prior and wavelet projections to estimate the regression function and all its mixed partial derivatives. We show that their posterior distributions contract to the truth optimally and adaptively under supremum-norm loss. The master theorem through tests with exponential errors used in Bayesian nonparametrics was not adequate to deal with this problem, and we developed a new idea such that posterior under the regression model is systematically reduced to a posterior arising from some quasi-white noise model, where the latter model greatly simplifies our rate calculations. Hence, this paper takes the first step in showing explicitly how one can translate results from white noise to regression model in a Bayesian setting.

math.ST↗

arXiv2018

Asymptotic Properties of Approximate Bayesian Computation

Approximate Bayesian computation allows for statistical analysis in models with intractable likelihoods. In this paper we consider the asymptotic behaviour of the posterior distribution obtained by this method. We give general results on the rate at which the posterior distribution concentrates on sets containing the true parameter, its limiting shape, and the asymptotic distribution of the posterior mean. These results hold under given rates for the tolerance used within the method, mild regularity conditions on the summary statistics, and a condition linked to identification of the true parameters. Implications for practitioners are discussed.

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