arXiv subjects

Emmanuel Caron

Publications and source records attributed to Emmanuel Caron.

6 recordsLinked to original sources

arXiv2020

A finite sample analysis of the benign overfitting phenomenon for ridge function estimation

Recent extensive numerical experiments in high scale machine learning have allowed to uncover a quite counterintuitive phase transition, as a function of the ratio between the sample size and the number of parameters in the model. As the number of parameters $p$ approaches the sample size $n$, the generalisation error increases, but surprisingly, it starts decreasing again past the threshold $p=n$. This phenomenon, brought to the theoretical community attention in \cite{belkin2019reconciling}, has been thoroughly investigated lately, more specifically for simpler models than deep neural networks, such as the linear model when the parameter is taken to be the minimum norm solution to the least-squares problem, firstly in the asymptotic regime when $p$ and $n$ tend to infinity, see e.g. \cite{hastie2019surprises}, and recently in the finite dimensional regime and more specifically for linear models \cite{bartlett2020benign}, \cite{tsigler2020benign}, \cite{lecue2022geometrical}. In the present paper, we propose a finite sample analysis of non-linear models of \textit{ridge} type, where we investigate the \textit{overparametrised regime} of the double descent phenomenon for both the \textit{estimation problem} and the \textit{prediction} problem. Our results provide a precise analysis of the distance of the best estimator from the true parameter as well as a generalisation bound which complements recent works of \cite{bartlett2020benign} and \cite{chinot2020benign}. Our analysis is based on tools closely related to the continuous Newton method \cite{neuberger2007continuous} and a refined quantitative analysis of the performance in prediction of the minimum $\ell_2$-norm solution.

stat.ML↗

arXiv2020

Gaussian linear model selection in a dependent context

In this paper, we study the nonparametric linear model, when the error process is a dependent Gaussian process. We focus on the estimation of the mean vector via a model selection approach. We first give the general theoretical form of the penalty function, ensuring that the penalized estimator among a collection of models satisfies an oracle inequality. Then we derive a penalty shape involving the spectral radius of the covariance matrix of the errors, which can be chosen proportional to the dimension when the error process is stationary and short range dependent. However, this penalty can be too rough in some cases, in particular when the error process is long range dependent. In a second part, we focus on the fixed-design regression model assuming that the error process is a stationary Gaussian process. We propose a model selection procedure in order to estimate the mean function via piecewise polynomials on a regular partition, when the error process is either short range dependent, long range dependent or anti-persistent. We present different kinds of penalties, depending on the memory of the process. For each case, an adaptive estimator is built, and the rates of convergence are computed. Thanks to several sets of simulations, we study the performance of these different penalties for all types of errors (short memory, long memory and anti-persistent errors). Finally, we give an application of our method to the well-known Nile data, which clearly shows that the type of dependence of the error process must be taken into account.

math.ST↗

arXiv2019

Linear regression with stationary errors : the R package slm

This paper introduces the R package slm which stands for Stationary Linear Models. The package contains a set of statistical procedures for linear regression in the general context where the error process is strictly stationary with short memory. We work in the setting of Hannan (1973), who proved the asymptotic normality of the (normalized) least squares estimators (LSE) under very mild conditions on the error process. We propose different ways to estimate the asymptotic covariance matrix of the LSE, and then to correct the type I error rates of the usual tests on the parameters (as well as confidence intervals). The procedures are evaluated through different sets of simulations, and two examples of real datasets are studied.

stat.AP↗

arXiv2018

Asymptotic distribution of least square estimators for linear models with dependent errors

In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan (1973), who proved a Central Limit Theorem for the usual least square estimator under general conditions on the design and on the error process. Whatever the design satisfying Hannan's conditions, we define an estimator of the covariance matrix and we prove its consistency under very mild conditions. As an application, we show how to modify the usual tests on the linear model in this dependent context, in such a way that the type-I error rate remains asymptotically correct, and we illustrate the performance of this procedure through different sets of simulations.

math.ST↗

arXiv2017

Asymptotic distribution of least squares estimators for linear models with dependent errors : regular designs

In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan, who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and on the error process. We show that for a large class of designs, the asymptotic covariance matrix is as simple as the independent and identically distributed case. We then estimate the covariance matrix using an estimator of the spectral density whose consistency is proved under very mild conditions.

math.ST↗

arXiv2012

Mixture model for designs in high dimensional regression and the LASSO

The LASSO is a recent technique for variable selection in the regression model \bean y & = & X\beta + z, \eean where $X\in \R^{n\times p}$ and $z$ is a centered gaussian i.i.d. noise vector $\mathcal N(0,\sigma^2I)$. The LASSO has been proved to achieve remarkable properties such as exact support recovery of sparse vectors when the columns are sufficently incoherent and low prediction error under even less stringent conditions. However, many matrices do not satisfy small coherence in practical applications and the LASSO estimator may thus suffer from what is known as the slow rate regime. The goal of the present paper is to study the LASSO from a slightly different perspective by proposing a mixture model for the design matrix which is able to capture in a natural way the potentially clustered nature of the columns in many practical situations. In this model, the columns of the design matrix are drawn from a Gaussian mixture model. Instead of requiring incoherence for the design matrix $X$, we only require incoherence of the much smaller matrix of the mixture's centers. Our main result states that $X\beta$ can be estimated with the same precision as for incoherent designs except for a correction term depending on the maximal variance in the mixture model.

math.ST↗