arXiv subjects

Adrien Saumard

Publications and source records attributed to Adrien Saumard.

23 records · Page 2Linked to original sources

arXiv2017

Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases

We investigate the optimality for model selection of the so-called slope heuristics, $V$-fold cross-validation and $V$-fold penalization in a heteroscedastic with random design regression context. We consider a new class of linear models that we call strongly localized bases and that generalize histograms, piecewise polynomials and compactly supported wavelets. We derive sharp oracle inequalities that prove the asymptotic optimality of the slope heuristics---when the optimal penalty shape is known---and $V$ -fold penalization. Furthermore, $V$-fold cross-validation seems to be suboptimal for a fixed value of $V$ since it recovers asymptotically the oracle learned from a sample size equal to $1-V^{-1}$ of the original amount of data. Our results are based on genuine concentration inequalities for the true and empirical excess risks that are of independent interest. We show in our experiments the good behavior of the slope heuristics for the selection of linear wavelet models. Furthermore, $V$-fold cross-validation and $V$-fold penalization have comparable efficiency.

math.ST↗

arXiv2016

On optimality of empirical risk minimization in linear aggregation

In the first part of this paper, we show that the small-ball condition, recently introduced by Mendelson (2015), may behave poorly for important classes of localized functions such as wavelets, piecewise polynomials or trigonometric polynomials, in particular leading to suboptimal estimates of the rate of convergence of ERM for the linear aggregation problem. In a second part, we recover optimal rates of covergence for the excess risk of ERM when the dictionary is made of trigonometric functions. Considering the bounded case, we derive the concentration of the excess risk around a single point, which is an information far more precise than the rate of convergence. In the general setting of a L2 noise, we finally refine the small ball argument by rightly selecting the directions we are looking at, in such a way that we obtain optimal rates of aggregation for the Fourier dictionary.

math.ST↗

arXiv2015

The Slope Heuristics in Heteroscedastic Regression

We consider the estimation of a regression function with random design and heteroscedastic noise in a nonparametric setting. More precisely, we address the problem of characterizing the optimal penalty when the regression function is estimated by using a penalized least-squares model selection method. In this context, we show the existence of a minimal penalty, defined to be the maximum level of penalization under which the model selection procedure totally misbehaves. The optimal penalty is shown to be twice the minimal one and to satisfy a non-asymptotic pathwise oracle inequality with leading constant almost one. Finally, the ideal penalty being unknown in general, we propose a hold-out penalization procedure and show that the latter is asymptotically optimal.

math.ST↗

arXiv2015

Optimal upper and lower bounds for the true and empirical excess risks in heteroscedastic least-squares regression

We consider the estimation of a bounded regression function with nonparametric heteroscedastic noise and random design. We study the true and empirical excess risks of the least-squares estimator on finite-dimensional vector spaces. We give upper and lower bounds on these quantities that are nonasymptotic and optimal to first order, allowing the dimension to depend on sample size. These bounds show the equivalence between the true and empirical excess risks when, among other things, the least-squares estimator is consistent in sup-norm with the projection of the regression function onto the considered model. Consistency in the sup-norm is then proved for suitable histogram models and more general models of piecewise polynomials that are endowed with a localized basis structure.

math.ST↗

arXiv2014

Log-concavity and strong log-concavity: a review

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

math.ST↗