arXiv subjects

Pierre C Bellec

Publications and source records attributed to Pierre C Bellec.

18 recordsLinked to original sources

arXiv2026

Generalization Error Estimation for Primal--Dual Algorithms in Non-Smooth Regression

This paper studies trajectory-wise estimation of generalization error for primal--dual algorithms in non-smooth regression. Motivating examples include $\ell_1$-penalized least absolute deviations regression and square-root Lasso regression, where the data-fitting loss is non-differentiable and existing risk estimators for gradient-type optimization paths do not apply directly. We develop a general recursive framework that includes the Chambolle--Pock algorithm and related primal--dual splitting methods. We estimate risk by correcting each in-sample fitted value with a weighted combination of past dual iterates. The ideal weights are Stein derivative contractions and depend on the design covariance. We construct replacement weights from observable derivative contractions of the fitted-signal trajectory, yielding a covariance-free, data-driven correction. For high-dimensional Gaussian designs and fixed finite iteration horizon, we prove finite-sample guarantees for both estimators. For square-root ridge, we further establish a matched-Gaussian universality result beyond Gaussian designs. Numerical experiments show that the proposed estimators accurately track the out-of-sample risk along finite optimization paths.

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arXiv2026

Observable adjustments in single-index models for regularized M-estimators

We consider observations $(X,y)$ from single index models with unknown link function, Gaussian covariates and a regularized M-estimator $\hatβ$ constructed from convex loss function and regularizer. In the regime where sample size $n$ and dimension $p$ are both increasing such that $p/n$ has a finite limit, the behavior of the empirical distribution of $\hatβ$ and the predicted values $X\hatβ$ has been previously characterized in a number of models: The empirical distributions are known to converge to proximal operators of the loss and penalty in a related Gaussian sequence model, which captures the interplay between ratio $p/n$, loss, regularization and the data generating process. This connection between$(\hatβ,X\hatβ)$ and the corresponding proximal operators require solving fixed-point equations that typically involve unobservable quantities such as the prior distribution on the index or the link function. This paper develops a different theory to describe the empirical distribution of $\hatβ$ and $X\hatβ$: Approximations of $(\hatβ,X\hatβ)$ in terms of proximal operators are provided that only involve observable adjustments. These proposed observable adjustments are data-driven, e.g., do not require prior knowledge of the index or the link function. These new adjustments yield confidence intervals for individual components of the index, as well as estimators of the correlation of $\hatβ$ with the index. The interplay between loss, regularization and the model is thus captured in a data-driven manner, without solving the fixed-point equations studied in previous works. The results apply to both strongly convex regularizers and unregularized M-estimation. Simulations are provided for the square and logistic loss in single index models including logistic regression and 1-bit compressed sensing with 20\% corrupted bits.

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arXiv2026

Simultaneous analysis of approximate leave-one-out cross-validation and mean-field inference

Approximate Leave-One-Out Cross-Validation (ALO-CV) is a method that has been proposed to estimate the generalization error of a regularized estimator in the high-dimensional regime where dimension and sample size are of the same order, the so-called ``proportional regime''. A new analysis is developed to derive the consistency of ALO-CV for non-differentiable regularizers under Gaussian covariates and strong convexity. Using a conditioning argument, the difference between the ALO-CV weights and their counterparts in mean-field inference is shown to be small. Combined with upper bounds between the mean-field inference estimate and the leave-one-out quantity, this provides a proof that ALO-CV approximates the leave-one-out quantity up to negligible error terms. Linear models with square loss, robust linear regression and single-index models are explicitly treated.

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arXiv2025

Optimizing over iid distributions and the Beat the Average game

A casino offers the following game. There are three cups each containing a die. You are being told that the dice in the cups are all the same, but possibly nonstandard. For a bet of \$1, the game master shakes all three cups and lets you choose one of them. You win \$2 if the die in your cup displays at least the average of the other two, and you lose otherwise. Is this game in your favor? If not, how should the casino design the dice to maximize their profit? This problem is a special case of the following more general question: given a measurable space $X$ and a bounded measurable function $f : X^n \to \R$, how large can the expectation of $f$ under probability measures of the form $μ^{\otimes n}$ be? We develop a general method to answer this kind of question. As an example application that is harder than the casino problem, we show that the maximal probability of the event $X_1 + X_2 + X_3 < 2 X_4$ for nonnegative iid random variables lies between $0.400695$ and $0.417$, where the upper bound is obtained by mixed integer linear programming. We conjecture the lower bound to be the exact value.

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arXiv2025

The noise barrier and the large signal bias of the Lasso and other convex estimators

Convex estimators such as the Lasso, the matrix Lasso and the group Lasso have been studied extensively in the last two decades, demonstrating great success in both theory and practice. Two quantities are introduced, the noise barrier and the large scale bias, that provides insights on the performance of these convex regularized estimators. It is now well understood that the Lasso achieves fast prediction rates, provided that the correlations of the design satisfy some Restricted Eigenvalue or Compatibility condition, and provided that the tuning parameter is large enough. Using the two quantities introduced in the paper, we show that the compatibility condition on the design matrix is actually unavoidable to achieve fast prediction rates with the Lasso. The Lasso must incur a loss due to the correlations of the design matrix, measured in terms of the compatibility constant. This results holds for any design matrix, any active subset of covariates, and any tuning parameter. It is now well known that the Lasso enjoys a dimension reduction property: the prediction error is of order $λ\sqrt k$ where $k$ is the sparsity; even if the ambient dimension $p$ is much larger than $k$. Such results require that the tuning parameters is greater than some universal threshold. We characterize sharp phase transitions for the tuning parameter of the Lasso around a critical threshold dependent on $k$. If $λ$ is equal or larger than this critical threshold, the Lasso is minimax over $k$-sparse target vectors. If $λ$ is equal or smaller than critical threshold, the Lasso incurs a loss of order $σ\sqrt k$ -- which corresponds to a model of size $k$ -- even if the target vector has fewer than $k$ nonzero coefficients. Remarkably, the lower bounds obtained in the paper also apply to random, data-driven tuning parameters. The results extend to convex penalties beyond the Lasso.

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arXiv2025

Derivatives and residual distribution of regularized M-estimators with application to adaptive tuning

This paper studies M-estimators with gradient-Lipschitz loss function regularized with convex penalty in linear models with Gaussian design matrix and arbitrary noise distribution. A practical example is the robust M-estimator constructed with the Huber loss and the Elastic-Net penalty and the noise distribution has heavy-tails. Our main contributions are three-fold. (i) We provide general formulae for the derivatives of regularized M-estimators $\hatβ(y,X)$ where differentiation is taken with respect to both $y$ and $X$; this reveals a simple differentiability structure shared by all convex regularized M-estimators. (ii) Using these derivatives, we characterize the distribution of the residual $r_i = y_i-x_i^\top\hatβ$ in the intermediate high-dimensional regime where dimension and sample size are of the same order. (iii) Motivated by the distribution of the residuals, we propose a novel adaptive criterion to select tuning parameters of regularized M-estimators. The criterion approximates the out-of-sample error up to an additive constant independent of the estimator, so that minimizing the criterion provides a proxy for minimizing the out-of-sample error. The proposed adaptive criterion does not require the knowledge of the noise distribution or of the covariance of the design. Simulated data confirms the theoretical findings, regarding both the distribution of the residuals and the success of the criterion as a proxy of the out-of-sample error. Finally our results reveal new relationships between the derivatives of $\hatβ(y,X)$ and the effective degrees of freedom of the M-estimator, which are of independent interest.

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arXiv2023

Out-of-sample error estimate for robust M-estimators with convex penalty

A generic out-of-sample error estimate is proposed for robust $M$-estimators regularized with a convex penalty in high-dimensional linear regression where $(X,y)$ is observed and $p,n$ are of the same order. If $ψ$ is the derivative of the robust data-fitting loss $ρ$, the estimate depends on the observed data only through the quantities $\hatψ= ψ(y-X\hatβ)$, $X^\top \hatψ$ and the derivatives $(\partial/\partial y) \hatψ$ and $(\partial/\partial y) X\hatβ$ for fixed $X$. The out-of-sample error estimate enjoys a relative error of order $n^{-1/2}$ in a linear model with Gaussian covariates and independent noise, either non-asymptotically when $p/n\le γ$ or asymptotically in the high-dimensional asymptotic regime $p/n\toγ'\in(0,\infty)$. General differentiable loss functions $ρ$ are allowed provided that $ψ=ρ'$ is 1-Lipschitz. The validity of the out-of-sample error estimate holds either under a strong convexity assumption, or for the $\ell_1$-penalized Huber M-estimator if the number of corrupted observations and sparsity of the true $β$ are bounded from above by $s_*n$ for some small enough constant $s_*\in(0,1)$ independent of $n,p$. For the square loss and in the absence of corruption in the response, the results additionally yield $n^{-1/2}$-consistent estimates of the noise variance and of the generalization error. This generalizes, to arbitrary convex penalty, estimates that were previously known for the Lasso.

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arXiv2022

Noise Covariance Estimation in Multi-Task High-dimensional Linear Models

This paper studies the multi-task high-dimensional linear regression models where the noise among different tasks is correlated, in the moderately high dimensional regime where sample size $n$ and dimension $p$ are of the same order. Our goal is to estimate the covariance matrix of the noise random vectors, or equivalently the correlation of the noise variables on any pair of two tasks. Treating the regression coefficients as a nuisance parameter, we leverage the multi-task elastic-net and multi-task lasso estimators to estimate the nuisance. By precisely understanding the bias of the squared residual matrix and by correcting this bias, we develop a novel estimator of the noise covariance that converges in Frobenius norm at the rate $n^{-1/2}$ when the covariates are Gaussian. This novel estimator is efficiently computable. Under suitable conditions, the proposed estimator of the noise covariance attains the same rate of convergence as the "oracle" estimator that knows in advance the regression coefficients of the multi-task model. The Frobenius error bounds obtained in this paper also illustrate the advantage of this new estimator compared to a method-of-moments estimator that does not attempt to estimate the nuisance. As a byproduct of our techniques, we obtain an estimate of the generalization error of the multi-task elastic-net and multi-task lasso estimators. Extensive simulation studies are carried out to illustrate the numerical performance of the proposed method.

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arXiv2021

De-biasing convex regularized estimators and interval estimation in linear models

New upper bounds are developed for the $L_2$ distance between $ξ/\text{Var}[ξ]^{1/2}$ and linear and quadratic functions of $z\sim N(0,I_n)$ for random variables of the form $ξ=bz^\top f(z) - \text{div} f(z)$. The linear approximation yields a central limit theorem when the squared norm of $f(z)$ dominates the squared Frobenius norm of $\nabla f(z)$ in expectation. Applications of this normal approximation are given for the asymptotic normality of de-biased estimators in linear regression with correlated design and convex penalty in the regime $p/n \le γ$ for constant $γ\in(0,{\infty})$. For the estimation of linear functions $\langle a_0,β\rangle$ of the unknown coefficient vector $β$, this analysis leads to asymptotic normality of the de-biased estimate for most normalized directions $a_0$, where ``most'' is quantified in a precise sense. This asymptotic normality holds for any convex penalty if $γ<1$ and for any strongly convex penalty if $γ\ge 1$. In particular the penalty needs not be separable or permutation invariant. By allowing arbitrary regularizers, the results vastly broaden the scope of applicability of de-biasing methodologies to obtain confidence intervals in high-dimensions. In the absence of strong convexity for $p>n$, asymptotic normality of the de-biased estimate is obtained for the Lasso and the group Lasso under additional conditions. For general convex penalties, our analysis also provides prediction and estimation error bounds of independent interest.

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arXiv2021

Chi-square and normal inference in high-dimensional multi-task regression

The paper proposes chi-square and normal inference methodologies for the unknown coefficient matrix $B^*$ of size $p\times T$ in a Multi-Task (MT) linear model with $p$ covariates, $T$ tasks and $n$ observations under a row-sparse assumption on $B^*$. The row-sparsity $s$, dimension $p$ and number of tasks $T$ are allowed to grow with $n$. In the high-dimensional regime $p\ggg n$, in order to leverage row-sparsity, the MT Lasso is considered. We build upon the MT Lasso with a de-biasing scheme to correct for the bias induced by the penalty. This scheme requires the introduction of a new data-driven object, coined the interaction matrix, that captures effective correlations between noise vector and residuals on different tasks. This matrix is psd, of size $T\times T$ and can be computed efficiently. The interaction matrix lets us derive asymptotic normal and $χ^2_T$ results under Gaussian design and $\frac{sT+s\log(p/s)}{n}\to0$ which corresponds to consistency in Frobenius norm. These asymptotic distribution results yield valid confidence intervals for single entries of $B^*$ and valid confidence ellipsoids for single rows of $B^*$, for both known and unknown design covariance $Σ$. While previous proposals in grouped-variables regression require row-sparsity $s\lesssim\sqrt n$ up to constants depending on $T$ and logarithmic factors in $n,p$, the de-biasing scheme using the interaction matrix provides confidence intervals and $χ^2_T$ confidence ellipsoids under the conditions ${\min(T^2,\log^8p)}/{n}\to 0$ and $$ \frac{sT+s\log(p/s)+\|Σ^{-1}e_j\|_0\log p}{n}\to0, \quad \frac{\min(s,\|Σ^{-1}e_j\|_0)}{\sqrt n} \sqrt{[T+\log(p/s)]\log p}\to 0, $$ allowing row-sparsity $s\ggg\sqrt n$ when $\|Σ^{-1}e_j\|_0 \sqrt T\lll \sqrt{n}$ up to logarithmic factors.

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arXiv2021

Asymptotic normality of robust $M$-estimators with convex penalty

This paper develops asymptotic normality results for individual coordinates of robust M-estimators with convex penalty in high-dimensions, where the dimension $p$ is at most of the same order as the sample size $n$, i.e, $p/n\leγ$ for some fixed constant $γ>0$. The asymptotic normality requires a bias correction and holds for most coordinates of the M-estimator for a large class of loss functions including the Huber loss and its smoothed versions regularized with a strongly convex penalty. The asymptotic variance that characterizes the width of the resulting confidence intervals is estimated with data-driven quantities. This estimate of the variance adapts automatically to low ($p/n\to0)$ or high ($p/n \le γ$) dimensions and does not involve the proximal operators seen in previous works on asymptotic normality of M-estimators. For the Huber loss, the estimated variance has a simple expression involving an effective degrees-of-freedom as well as an effective sample size. The case of the Huber loss with Elastic-Net penalty is studied in details and a simulation study confirms the theoretical findings. The asymptotic normality results follow from Stein formulae for high-dimensional random vectors on the sphere developed in the paper which are of independent interest.

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arXiv2020

First order expansion of convex regularized estimators

We consider first order expansions of convex penalized estimators in high-dimensional regression problems with random designs. Our setting includes linear regression and logistic regression as special cases. For a given penalty function $h$ and the corresponding penalized estimator $\hatβ$, we construct a quantity $η$, the first order expansion of $\hatβ$, such that the distance between $\hatβ$ and $η$ is an order of magnitude smaller than the estimation error $\|\hatβ - β^*\|$. In this sense, the first order expansion $η$ can be thought of as a generalization of influence functions from the mathematical statistics literature to regularized estimators in high-dimensions. Such first order expansion implies that the risk of $\hatβ$ is asymptotically the same as the risk of $η$ which leads to a precise characterization of the MSE of $\hatβ$; this characterization takes a particularly simple form for isotropic design. Such first order expansion also leads to inference results based on $\hatβ$. We provide sufficient conditions for the existence of such first order expansion for three regularizers: the Lasso in its constrained form, the lasso in its penalized form, and the Group-Lasso. The results apply to general loss functions under some conditions and those conditions are satisfied for the squared loss in linear regression and for the logistic loss in the logistic model.

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arXiv2020

Second order Stein: SURE for SURE and other applications in high-dimensional inference

Stein's formula states that a random variable of the form $z^\top f(z) - \text{div} f(z)$ is mean-zero for functions $f$ with integrable gradient. Here, $\text{div} f$ is the divergence of the function $f$ and $z$ is a standard normal vector. This paper aims to propose a Second Order Stein formula to characterize the variance of such random variables for all functions $f(z)$ with square integrable gradient, and to demonstrate the usefulness of this formula in various applications. In the Gaussian sequence model, a consequence of Stein's formula is Stein's Unbiased Risk Estimate (SURE), an unbiased estimate of the mean squared risk for almost any estimator $\hatμ$ of the unknown mean. A first application of the Second Order Stein formula is an Unbiased Risk Estimate for SURE itself (SURE for SURE): an unbiased estimate {providing} information about the squared distance between SURE and the squared estimation error of $\hatμ$. SURE for SURE has a simple form as a function of the data and is applicable to all $\hatμ$ with square integrable gradient, e.g. the Lasso and the Elastic Net. In addition to SURE for SURE, the following applications are developed: (1) Upper bounds on the risk of SURE when the estimation target is the mean squared error; (2) Confidence regions based on SURE; (3) Oracle inequalities satisfied by SURE-tuned estimates; (4) An upper bound on the variance of the size of the model selected by the Lasso; (5) Explicit expressions of SURE for SURE for the Lasso and the Elastic-Net; (6) In the linear model, a general semi-parametric scheme to de-bias a differentiable initial estimator for inference of a low-dimensional projection of the unknown $β$, with a characterization of the variance after de-biasing; and (7) An accuracy analysis of a Gaussian Monte Carlo scheme to approximate the divergence of functions $f: R^n\to R^n$.

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arXiv2019

The cost-free nature of optimally tuning Tikhonov regularizers and other ordered smoothers

We consider the problem of selecting the best estimator among a family of Tikhonov regularized estimators, or, alternatively, to select a linear combination of these regularizers that is as good as the best regularizer in the family. Our theory reveals that if the Tikhonov regularizers share the same penalty matrix with different tuning parameters, a convex procedure based on $Q$-aggregation achieves the mean square error of the best estimator, up to a small error term no larger than $Cσ^2$, where $σ^2$ is the noise level and $C>0$ is an absolute constant. Remarkably, the error term does not depend on the penalty matrix or the number of estimators as long as they share the same penalty matrix, i.e., it applies to any grid of tuning parameters, no matter how large the cardinality of the grid is. This reveals the surprising "cost-free" nature of optimally tuning Tikhonov regularizers, in striking contrast with the existing literature on aggregation of estimators where one typically has to pay a cost of $σ^2\log(M)$ where $M$ is the number of estimators in the family. The result holds, more generally, for any family of ordered linear smoothers. This encompasses Ridge regression as well as Principal Component Regression. The result is extended to the problem of tuning Tikhonov regularizers with different penalty matrices.

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arXiv2019

Concentration of quadratic forms under a Bernstein moment assumption

A concentration result for quadratic form of independent subgaussian random variables is derived. If the moments of the random variables satisfy a "Bernstein condition", then the variance term of the Hanson-Wright inequality can be improved. The Bernstein condition is satisfied, for instance, by all log-concave subgaussian distributions.

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arXiv2017

A sharp oracle inequality for Graph-Slope

Following recent success on the analysis of the Slope estimator, we provide a sharp oracle inequality in term of prediction error for Graph-Slope, a generalization of Slope to signals observed over a graph. In addition to improving upon best results obtained so far for the Total Variation denoiser (also referred to as Graph-Lasso or Generalized Lasso), we propose an efficient algorithm to compute Graph-Slope. The proposed algorithm is obtained by applying the forward-backward method to the dual formulation of the Graph-Slope optimization problem. We also provide experiments showing the interest of the method.

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arXiv2017

Optimistic lower bounds for convex regularized least-squares

Minimax lower bounds are pessimistic in nature: for any given estimator, minimax lower bounds yield the existence of a worst-case target vector $β^*_{worst}$ for which the prediction error of the given estimator is bounded from below. However, minimax lower bounds shed no light on the prediction error of the given estimator for target vectors different than $β^*_{worst}$. A characterization of the prediction error of any convex regularized least-squares is given. This characterization provide both a lower bound and an upper bound on the prediction error. This produces lower bounds that are applicable for any target vector and not only for a single, worst-case $β^*_{worst}$. Finally, these lower and upper bounds on the prediction error are applied to the Lasso is sparse linear regression. We obtain a lower bound involving the compatibility constant for any tuning parameter, matching upper and lower bounds for the universal choice of the tuning parameter, and a lower bound for the Lasso with small tuning parameter.

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arXiv2017

Localized Gaussian width of $M$-convex hulls with applications to Lasso and convex aggregation

Upper and lower bounds are derived for the Gaussian mean width of the intersection of a convex hull of $M$ points with an Euclidean ball of a given radius. The upper bound holds for any collection of extreme point bounded in Euclidean norm. The upper bound and the lower bound match up to a multiplicative constant whenever the extreme points satisfy a one sided Restricted Isometry Property. This bound is then applied to study the Lasso estimator in fixed-design regression, the Empirical Risk Minimizer in the anisotropic persistence problem, and the convex aggregation problem in density estimation.

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