Séance du 17 avril 2023

Séance organisée par Alain Célisse et Céline Duval 

Lieu : IHP,  amphi Hermite

14.00 :  Déborah Sulem  (Pompeu Fabra)

Titre :  Scalable Variational Bayes methods for Hawkes processes

Résumé : Multivariate Hawkes processes are temporal point processes extensively applied to model event data with dependence on past occurrences and interaction phenomena, e.g., neuronal spike trains, online messages, and financial transactions. In the nonparametric setting, learning the temporal dependence structure of Hawkes processes is often a computationally expensive task, all the more with Bayesian estimation methods. In the generalised nonlinear Hawkes model, the posterior distribution is non-conjugate and doubly intractable, and existing Monte-Carlo Markov Chain methods are often slow and not scalable to high-dimensional processes in practice. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed. In this work, we unify existing variational Bayes inference approaches under a general framework, that we theoretically analyse under easily verifiable conditions on the prior, the variational class, and the model. Then, in the context of the popular sigmoid Hawkes model, we design adaptive and sparsity-inducing mean-field variational methods. In particular, we propose a two-step algorithm based on a thresholding heuristic to select the connectivity graph parameter of the Hawkes model. Through an extensive set of numerical simulations, we demonstrate that our approach enjoys several benefits: it is computationally efficient, can reduce the dimensionality of the problem by selecting the graph parameter, and is able to adapt to the smoothness of the underlying parameter.

15.00 :  Claire Boyer (Sorbonne Université)

Titre :  Is interpolation benign for random forest regression? 

Résumé : Statistical wisdom suggests that very complex models, interpolating training data, will be poor at predicting unseen examples. Yet, this aphorism has been recently challenged by the identification of benign overfitting regimes, specially studied in the case of parametric models: generalization capabilities may be preserved despite model high complexity. While it is widely known that fully-grown decision trees interpolate and, in turn, have bad predictive performances, the same behavior is yet to be analyzed for Random Forests (RF). In this paper, we study the trade-off between interpolation and consistency for several types of RF algorithms. Theoretically, we prove that interpolation regimes and consistency cannot be achieved simultaneously for several non-adaptive RF. Since adaptivity seems to be the cornerstone to bring together interpolation and consistency, we study interpolating Median RF which are proved to be consistent in the interpolating regime. This is the first result conciliating interpolation and consistency for RF, highlighting that the averaging effect introduced by feature randomization is a key mechanism, sufficient to ensure the consistency in the interpolation regime and beyond. Numerical experiments show that Breiman's RF are consistent while exactly interpolating, when no bootstrap step is involved. We theoretically control the size of the interpolation area, which converges fast enough to zero, giving a necessary condition for exact interpolation and consistency to occur in conjunction. 

16.00 :  Victor-Emmanuel Brunel (CREST)

Titre : Statistical properties of barycenters in metric spaces

Résumé :  Barycenters, aka Fréchet means, offer a natural extension of averages from linear spaces to general metric spaces. Just as in Euclidean spaces, limit theorems (such as laws of large numbers and central limit theorems) are well known under fairly general assumptions. In this talk, after a brief introduction of barycenters in metric spaces, I will present a framework - that of geodesic spaces with non-positive curvature - where non-asymptotic guarantees can be proven. This talk is inspired from a recent work in collaboration with Jordan Serres (ENSAE).