arXiv subjects

Antoine Godichon-Baggioni

Publications and source records attributed to Antoine Godichon-Baggioni.

29 records · Page 2Linked to original sources

arXiv2023

Online stochastic Newton methods for estimating the geometric median and applications

In the context of large samples, a small number of individuals might spoil basic statistical indicators like the mean. It is difficult to detect automatically these atypical individuals, and an alternative strategy is using robust approaches. This paper focuses on estimating the geometric median of a random variable, which is a robust indicator of central tendency. In order to deal with large samples of data arriving sequentially, online stochastic Newton algorithms for estimating the geometric median are introduced and we give their rates of convergence. Since estimates of the median and those of the Hessian matrix can be recursively updated, we also determine confidences intervals of the median in any designated direction and perform online statistical tests.

stat.ML↗

arXiv2023

Non asymptotic analysis of Adaptive stochastic gradient algorithms and applications

In stochastic optimization, a common tool to deal sequentially with large sample is to consider the well-known stochastic gradient algorithm. Nevertheless, since the stepsequence is the same for each direction, this can lead to bad results in practice in case of ill-conditionned problem. To overcome this, adaptive gradient algorithms such that Adagrad or Stochastic Newton algorithms should be prefered. This paper is devoted to the non asymptotic analyis of these adaptive gradient algorithms for strongly convex objective. All the theoretical results will be adapted to linear regression and regularized generalized linear model for both Adagrad and Stochastic Newton algorithms.

math.OC↗

arXiv2022

A robust model-based clustering based on the geometric median and the Median Covariation Matrix

Grouping observations into homogeneous groups is a recurrent task in statistical data analysis. We consider Gaussian Mixture Models, which are the most famous parametric model-based clustering method. We propose a new robust approach for model-based clustering, which consists in a modification of the EM algorithm (more specifically, the M-step) by replacing the estimates of the mean and the variance by robust versions based on the median and the median covariation matrix. All the proposed methods are available in the R package RGMM accessible on CRAN.

stat.ME↗

arXiv2022

Lp and almost sure rates of convergence of averaged stochastic gradient algorithms: locally strongly convex objective

An usual problem in statistics consists in estimating the minimizer of a convex function. When we have to deal with large samples taking values in high dimensional spaces, stochastic gradient algorithms and their averaged versions are efficient candidates. Indeed, (1) they do not need too much computational efforts, (2) they do not need to store all the data, which is crucial when we deal with big data, (3) they allow to simply update the estimates, which is important when data arrive sequentially. The aim of this work is to give asymptotic and non asymptotic rates of convergence of stochastic gradient estimates as well as of their averaged versions when the function we would like to minimize is only locally strongly convex.

math.ST↗

arXiv2020

An efficient Averaged Stochastic Gauss-Newton algorithm for estimating parameters of non linear regressions models

Non linear regression models are a standard tool for modeling real phenomena, with several applications in machine learning, ecology, econometry... Estimating the parameters of the model has garnered a lot of attention during many years. We focus here on a recursive method for estimating parameters of non linear regressions. Indeed, these kinds of methods, whose most famous are probably the stochastic gradient algorithm and its averaged version, enable to deal efficiently with massive data arriving sequentially. Nevertheless, they can be, in practice, very sensitive to the case where the eigen-values of the Hessian of the functional we would like to minimize are at different scales. To avoid this problem, we first introduce an online Stochastic Gauss-Newton algorithm. In order to improve the estimates behavior in case of bad initialization, we also introduce a new Averaged Stochastic Gauss-Newton algorithm and prove its asymptotic efficiency.

math.ST↗

arXiv2019

An efficient stochastic Newton algorithm for parameter estimation in logistic regressions

Logistic regression is a well-known statistical model which is commonly used in the situation where the output is a binary random variable. It has a wide range of applications including machine learning, public health, social sciences, ecology and econometry. In order to estimate the unknown parameters of logistic regression with data streams arriving sequentially and at high speed, we focus our attention on a recursive stochastic algorithm. More precisely, we investigate the asymptotic behavior of a new stochastic Newton algorithm. It enables to easily update the estimates when the data arrive sequentially and to have research steps in all directions. We establish the almost sure convergence of our stochastic Newton algorithm as well as its asymptotic normality. All our theoretical results are illustrated by numerical experiments.

math.ST↗

arXiv2017

On the rates of convergence of Parallelized Averaged Stochastic Gradient Algorithms

The growing interest for high dimensional and functional data analysis led in the last decade to an important research developing a consequent amount of techniques. Parallelized algorithms, which consist in distributing and treat the data into different machines, for example, are a good answer to deal with large samples taking values in high dimensional spaces. We introduce here a parallelized averaged stochastic gradient algorithm, which enables to treat efficiently and recursively the data, and so, without taking care if the distribution of the data into the machines is uniform. The rate of convergence in quadratic mean as well as the asymptotic normality of the parallelized estimates are given, for strongly and locally strongly convex objectives.

math.ST↗

arXiv2017

Online estimation of the asymptotic variance for averaged stochastic gradient algorithms

Stochastic gradient algorithms are more and more studied since they can deal efficiently and online with large samples in high dimensional spaces. In this paper, we first establish a Central Limit Theorem for these estimates as well as for their averaged version in general Hilbert spaces. Moreover, since having the asymptotic normality of estimates is often unusable without an estimation of the asymptotic variance, we introduce a new recursive algorithm for estimating this last one, and we establish its almost sure rate of convergence as well as its rate of convergence in quadratic mean. Finally, two examples consisting in estimating the parameters of the logistic regression and estimating geometric quantiles are given.

math.ST↗

arXiv2017

Clustering transformed compositional data using K-means, with applications in gene expression and bicycle sharing system data

Although there is no shortage of clustering algorithms proposed in the literature, the question of the most relevant strategy for clustering compositional data (i.e., data made up of profiles, whose rows belong to the simplex) remains largely unexplored in cases where the observed value of an observation is equal or close to zero for one or more samples. This work is motivated by the analysis of two sets of compositional data, both focused on the categorization of profiles but arising from considerably different applications: (1) identifying groups of co-expressed genes from high-throughput RNA sequencing data, in which a given gene may be completely silent in one or more experimental conditions; and (2) finding patterns in the usage of stations over the course of one week in the Velib' bicycle sharing system in Paris, France. For both of these applications, we focus on the use of appropriately chosen data transformations, including the Centered Log Ratio and a novel extension we propose called the Log Centered Log Ratio, in conjunction with the K-means algorithm. We use a nonasymptotic penalized criterion, whose penalty is calibrated with the slope heuristics, to select the number of clusters present in the data. Finally, we illustrate the performance of this clustering strategy, which is implemented in the Bioconductor package coseq, on both the gene expression and bicycle sharing system data.

math.ST↗

arXiv2016

Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis

The geometric median covariation matrix is a robust multivariate indicator of dispersion which can be extended without any difficulty to functional data. We define estimators, based on recursive algorithms, that can be simply updated at each new observation and are able to deal rapidly with large samples of high dimensional data without being obliged to store all the data in memory. Asymptotic convergence properties of the recursive algorithms are studied under weak conditions. The computation of the principal components can also be performed online and this approach can be useful for online outlier detection. A simulation study clearly shows that this robust indicator is a competitive alternative to minimum covariance determinant when the dimension of the data is small and robust principal components analysis based on projection pursuit and spherical projections for high dimension data. An illustration on a large sample and high dimensional dataset consisting of individual TV audiences measured at a minute scale over a period of 24 hours confirms the interest of considering the robust principal components analysis based on the median covariation matrix. All studied algorithms are available in the R package Gmedian on CRAN.

math.ST↗

arXiv2016

An averaged projected Robbins-Monro algorithm for estimating the parameters of a truncated spherical distribution

The objective of this work is to propose a new algorithm to fit a sphere on a noisy 3D point cloud distributed around a complete or a truncated sphere. More precisely, we introduce a projected Robbins-Monro algorithm and its averaged version for estimating the center and the radius of the sphere. We give asymptotic results such as the almost sure convergence of these algorithms as well as the asymptotic normality of the averaged algorithm. Furthermore, some non-asymptotic results will be given, such as the rates of convergence in quadratic mean. Some numerical experiments show the efficiency of the proposed algorithm on simulated data for small to moderate sample sizes.

math.ST↗