Prochainement

Jeudi 21 novembre 14:00-15:00 Christophe Denis (LAMA - UPEM)
Minimax semi-supervised confidence set for multi-class classification

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Résumé : Multiclass classification problems such as image annotation can involve a large number of classes.
In this context, confusion between classes can occur, and a single label classification may fail. In this talk, I will present a general device to build a confidence set classifier, instead of a single label classifier.
In our framework the goal is to build the best confidence set classifier having a given expected size and the attractive feature of our approach is its semi-supervised nature - the construction of the confidence set classifier takes advantage of unlabeled data.
Our study of the minimax rates of convergence under the combination of the margin and non parametric assumptions reveals that there is no supervised method that outperforms the semi-supervised estimator proposed in this work.
To further highlight the fundamental difference of supervised and semi-supervised methods, we establish that the best achievable rate for any supervised method is n^-1/2, even if the margin assumption is extremely favourable.
On the contrary, by using a sufficiently large unlabelled sample we are able to significantly improve this rate.

Minimax semi-supervised confidence set for multi-class classification  Version PDF
Jeudi 21 novembre 15:45-16:45 Alex Karrila (IHES)
On multiple SLE type scaling limits

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Résumé : Schramm-Loewner evolution (SLE) type curves are conformally invariant random curves, known or conjectured to describe the scaling limits of random interfaces in many critical planar lattice models. A particularly interesting variant are multiple SLE curve collections, which explicitly connect SLEs to Conformal field theory, the physics description of such scaling limits. We recall these notions, sketch proofs realizing multiple SLEs in scaling limits of lattice models, and discuss their further consequences. The talk is mainly based on [arXiv:1903.10354] and [arXiv:1810.05608].

On multiple SLE type scaling limits  Version PDF
Jeudi 5 décembre 15:45-16:45 Vincent Bansaye (CMAP, Ecole Polytechnique.)
TBA
Jeudi 12 décembre 15:45-16:45 Djalil Chafaï (CEREMADE, Université Paris-Dauphine)
TBA

Passés

Jeudi 14 novembre 15:45-16:45 Giovanni Conforti (CMAP, Ecole Polytechnique.)
A large deviations perspective on functional inequalities

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Résumé : Functional inequalities are powerful tools to quantify the rate of convergence to equilibrium of Markov processes or to study the concentration of measure phenomenon.The aim of this talk is to explore a novel class of functional inequalities that has been recently obtained in connection with the Schrödinger problem and to show how they can be applied to obtain quantitative rates of convergence to equilibrium for (mean field) stochastic control problems. Leveraging the large deviations interpretation of the Schrödinger problem, we will also present some ideas that allow to define an abstract notion of transport inequality associated with a large deviation principle and test this definition on some model examples.

A large deviations perspective on functional inequalities  Version PDF
Jeudi 7 novembre 15:45-16:45 Xinxin Chen (ICJ, Université Claude Bernard, Lyon)
Lower and moderate deviation for maximums of branching random walk and branching Brownian motion

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Résumé : For a supercritical branching random walk on real line, it is proved by Bramson (1983) and Aïdékon (2013) that M_n, the maximal position at time n, shifted by its median m_n=x^* n-\frac32\theta^* \log n+\Theta(1), converges in law under some mild condition. We study the lower and moderate deviation for this convergence. Moreover, for branching Brownian motion, we study the process conditioned on small maximum. This is based on joint works with Hui He and Bastien Mallein.

Lower and moderate deviation for maximums of branching random walk and branching Brownian motion  Version PDF
Jeudi 7 novembre 14:00-15:00 Stéphane Gaïffas 
An improper estimator with optimal excess risk in misspecified density estimation and logistic regression

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Lieu : 3L15 - IMO

Résumé : (travail en collaboration avec Jaouad Mourtada et Erwan Scornet)Retour ligne automatique
We introduce a new procedure called SMP (Sample Minimax Predictor) for predictive conditional density estimation, which satisfies a general excess risk bound under logarithmic loss. This bound remains valid in the misspecified case, and scales as d / n in several cases, where d is the model dimension and n the sample size.Retour ligne automatique
In particular, and contrary to the maximum likelihood, the performance of this procedure does not significantly degrade under model misspecification.
We deduce a minimax procedure for misspecified density estimation in logistic regression, with a sharp excess risk of d / n + o(1/n), addressing an open problem by Kotlowski and Grunwald (2011).Retour ligne automatique
For logistic regression, the predictions of SMP come at the cost of two logistic regressions, hence are easier to compute than the approaches based on Bayesian predictive posteriors, which require posterior sampling instead of optimization.
From a theoretical point of view, SMP bypasses existing lower bounds for proper estimators, which return a conditional distribution that belongs to the logistic model. Results from Hazan et al (2014) (see also Bach and Moulines, 2013) imply that the excess risk rate of such procedures is either slow O (1 / \sqrtn) or exhibits an exponential dependence on the scale of the covariates for some worst-case distributions. It was shown recently by Foster et al (2018) that one can achieve a fast rate O(d \log n / n) using a mixture of Bayesian predictive posteriors. A Ridge-regularized variant of SMP also satisfies a fast rate, and therefore provides a computationally appealing alternative to the approach of Foster et al (2018).

An improper estimator with optimal excess risk in misspecified density estimation and logistic regression  Version PDF
Jeudi 24 octobre 14:00-15:00  
La séance est reportée au 7 novembre

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Lieu : 3L15 - IMO

Résumé : Stéphane Gaïffas
(travail en collaboration avec Jaouad Mourtada et Erwan Scornet)
We introduce a new procedure called SMP (Sample Minimax Predictor) for predictive conditional density estimation, which satisfies a general excess risk bound under logarithmic loss. This bound remains valid in the misspecified case, and scales as d / n in several cases, where d is the model dimension and n the sample size.
In particular, and contrary to the maximum likelihood, the performance of this procedure does not significantly degrade under model misspecification.
We deduce a minimax procedure for misspecified density estimation in logistic regression, with a sharp excess risk of d / n + o(1/n), addressing an open problem by Kotlowski and Grunwald (2011).
For logistic regression, the predictions of SMP come at the cost of two logistic regressions, hence are easier to compute than the approaches based on Bayesian predictive posteriors, which require posterior sampling instead of optimization.
From a theoretical point of view, SMP bypasses existing lower bounds for proper estimators, which return a conditional distribution that belongs to the logistic model. Results from Hazan et al (2014) (see also Bach and Moulines, 2013) imply that the excess risk rate of such procedures is either slow O (1 / \sqrtn) or exhibits an exponential dependence on the scale of the covariates for some worst-case distributions. It was shown recently by Foster et al (2018) that one can achieve a fast rate O(d \log n / n) using a mixture of Bayesian predictive posteriors. A Ridge-regularized variant of SMP also satisfies a fast rate, and therefore provides a computationally appealing alternative to the approach of Foster et al (2018).

Notes de dernières minutes : Attention, cette séance est reportée au 7 novembre

La séance est reportée au 7 novembre  Version PDF
Jeudi 17 octobre 15:45-16:45 Joseph Najnudel (University of Bristol)
TBA
Jeudi 10 octobre 15:45-16:45 Kilian Raschel (Denis Poisson (Université de Tours))
Processus aléatoires confinés et théorie de Galois des équations aux différences

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Résumé : Certains modèles de marches aléatoires ou mouvements browniens confinés dans des cônes peuvent s’étudier au moyen d’équations fonctionnelles pour les fonctions génératrices ou transformées de Laplace sous-jacentes. Cette approche s’applique à un nombre varié de situations, et donne accès à l’étude des probabilités d’atteindre les bords des cônes (mentionnons ici un lien avec la biologie des populations, où ces probabilités d’atteinte s’interprètent comme des probabilités d’extinction), à l’énumération des marches en combinatoire, aux distributions stationnaires de browniens réfléchis, etc. Il est assez aisé de reformuler ces équations fonctionnelles en termes d’équations aux différences, par exemple f(q*s) – f(s) = g(s), où f est la fonction inconnue (typiquement une fonction génératrice), tandis que g et q sont respectivement une fonction et un paramètre connus, dépendant du modèle. Des outils provenant de la théorie des équations aux différences (et leur théorie de Galois associée) se révèlent alors parfaitement adaptés, en particulier pour caractériser la nature algébrique de la solution, voire même pour la calculer ! Dans cet exposé nous rassemblerons plusieurs exemples : nous commencerons par évoquer le cas de l’énumération des chemins dans le quadrant, pour lequel des travaux récents de Dreyfus, Hardouin, Roques et Singer caractérisent la transcendance différentielle des fonctions génératrices. Nous nous intéresserons également au mouvement brownien réfléchi dans des cônes planaires et présenterons un travail commun avec Bousquet-Mélou, Elvey Price, Franceschi et Hardouin, donnant une caractérisation complète des transformées de Laplace des distributions stationnaires. Nous terminerons en montrant quelques problèmes ouverts.

Processus aléatoires confinés et théorie de Galois des équations aux différences  Version PDF
Jeudi 10 octobre 14:00-15:00 Sophie Donnet (INRA)
Bayesian inference for network Poisson models

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Résumé : This work is motivated by the analysis of ecological interaction networks. Poisson stochastic blockmodels are widely used in this field to decipher the structure that underlies a weighted network, while accounting for covariate effects. Efficient algorithms based on variational approximations exist for frequentist inference, but without statistical guaranties as for the resulting estimates. In absence of variational Bayes estimates, we show that a good proxy of the posterior distribution can be straightforwardly derived from the frequentist variational estimation procedure, using a Laplace approximation. We use this proxy to sample from the true posterior distribution via a sequential Monte-Carlo algorithm. As shown in the simulation study, the efficiency of the posterior sampling is greatly improved by the accuracy of the approximate posterior distribution. The proposed procedure can be easily extended to other latent variable models. We use this methodology to assess the influence of available covariates on the organization of two ecological networks, as well as the existence of a residual interaction structure
Joint work with Stéphane Robin.

Bayesian inference for network Poisson models  Version PDF