Related papers: Minimum $\ell_{1}$-norm interpolators: Precise asy…

We study the generalization capability of nearly-interpolating linear regressors: $\boldsymbol{\beta}$'s whose training error $\tau$ is positive but small, i.e., below the noise floor. Under a random matrix theoretic assumption on the data…

It is now well understood that $\ell_1$ minimization algorithm is able to recover sparse signals from incomplete measurements [2], [1], [3] and sharp recoverable sparsity thresholds have also been obtained for the $\ell_1$ minimization…

In this paper we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted $\ell_1$ minimization over that sparsity model. In particular, we focus on a model where the entries of the unknown vector fall into…

In energy-efficient schemes, finding the optimal size of deep learning models is very important and has a broad impact. Meanwhile, recent studies have reported an unexpected phenomenon, the sparse double descent: as the model's sparsity…

Consider the standard Gaussian linear regression model $Y=X\theta+\epsilon$, where $Y\in R^n$ is a response vector and $ X\in R^{n*p}$ is a design matrix. Numerous work have been devoted to building efficient estimators of $\theta$ when $p$…

In recent years, there has been a significant growth in research focusing on minimum $\ell_2$ norm (ridgeless) interpolation least squares estimators. However, the majority of these analyses have been limited to an unrealistic regression…

We consider the compressed sensing problem, where the object $x_0 \in \bR^N$ is to be recovered from incomplete measurements $y = Ax_0 + z$; here the sensing matrix $A$ is an $n \times N$ random matrix with iid Gaussian entries and $n < N$.…

We introduce estimation and test procedures through divergence minimization for models satisfying linear constraints with unknown parameter. Several statistical examples and motivations are given. These procedures extend the empirical…

We study the large sample properties of sparse M-estimators in the presence of pseudo-observations. Our framework covers a broad class of semi-parametric copula models, for which the marginal distributions are unknown and replaced by their…

Overparametrized neural networks tend to perfectly fit noisy training data yet generalize well on test data. Inspired by this empirical observation, recent work has sought to understand this phenomenon of benign overfitting or harmless…

Sparse linear regression -- finding an unknown vector from linear measurements -- is now known to be possible with fewer samples than variables, via methods like the LASSO. We consider the multiple sparse linear regression problem, where…

In the linear regression model, the minimum l2-norm interpolant estimator has received much attention since it was proved to be consistent even though it fits noisy data perfectly under some condition on the covariance matrix $\Sigma$ of…

We study the generalization error of functions that interpolate prescribed data points and are selected by minimizing a weighted norm. Under natural and general conditions, we prove that both the interpolants and their generalization errors…

The over-parameterized models attract much attention in the era of data science and deep learning. It is empirically observed that although these models, e.g. deep neural networks, over-fit the training data, they can still achieve small…

The rapid recent progress in machine learning (ML) has raised a number of scientific questions that challenge the longstanding dogma of the field. One of the most important riddles is the good empirical generalization of overparameterized…

Adversarial training has been proposed to protect machine learning models against adversarial attacks. This paper focuses on adversarial training under $\ell_\infty$-perturbation, which has recently attracted much research attention. The…

Classical optimization theory requires a small step-size for gradient-based methods to converge. Nevertheless, recent findings challenge the traditional idea by empirically demonstrating Gradient Descent (GD) converges even when the…

We study least squares linear regression over $N$ uncorrelated Gaussian features that are selected in order of decreasing variance. When the number of selected features $p$ is at most the sample size $n$, the estimator under consideration…

Modern machine learning models are often trained in a setting where the number of parameters exceeds the number of training samples. To understand the implicit bias of gradient descent in such overparameterized models, prior work has…

We study the problem of learning causal models from observational data through the lens of interpolation and its counterpart -- regularization. A large volume of recent theoretical, as well as empirical work, suggests that, in highly…