Many real-world data sets can be presented in the form of a matrix whose entries correspond to the interaction between two entities of different natures (number of times a web user visits a web page, a student's grade in a subject, a patient's rating of a doctor, etc.). We assume in this paper that the mentioned interaction is determined by unobservable latent variables describing each entity. Our objective is to estimate the conditional expectation of the data matrix given the unobservable variables. This is presented as a problem of estimation of a bivariate function referred to as graphon. We study the cases of piecewise constant and H\"older-continuous graphons. We establish finite sample risk bounds for the least squares estimator and the exponentially weighted aggregate. These bounds highlight the dependence of the estimation error on the size of the data set, the maximum intensity of the interactions, and the level of noise. As the analyzed least-squares estimator is intractable, we propose an adaptation of Lloyd's alternating minimization algorithm to compute an approximation of the least-squares estimator. Finally, we present numerical experiments in order to illustrate the empirical performance of the graphon estimator on synthetic data sets.

We study the problem of sampling from a probability distribution on $\mathbb R^p$ defined via a convex and smooth potential function. We consider a continuous-time diffusion-type process, termed Penalized Langevin dynamics (PLD), the drift of which is the negative gradient of the potential plus a linear penalty that vanishes when time goes to infinity. An upper bound on the Wasserstein-2 distance between the distribution of the PLD at time $t$ and the target is established. This upper bound highlights the influence of the speed of decay of the penalty on the accuracy of the approximation. As a consequence, considering the low-temperature limit we infer a new nonasymptotic guarantee of convergence of the penalized gradient flow for the optimization problem.

In this paper, we revisit the recently established theoretical guarantees for the convergence of the Langevin Monte Carlo algorithm of sampling from a smooth and (strongly) log-concave density. We improve, in terms of constants, the existing results when the accuracy of sampling is measured in the Wasserstein distance and provide further insights on relations between, on the one hand, the Langevin Monte Carlo for sampling and, on the other hand, the gradient descent for optimization. More importantly, we establish non-asymptotic guarantees for the accuracy of a version of the Langevin Monte Carlo algorithm that is based on inaccurate evaluations of the gradient. Finally, we propose a variable-step version of the Langevin Monte Carlo algorithm that has two advantages. First, its step-sizes are independent of the target accuracy and, second, its rate provides a logarithmic improvement over the constant-step Langevin Monte Carlo algorithm.

Although the Lasso has been extensively studied, the relationship between its prediction performance and the correlations of the covariates is not fully understood. In this paper, we give new insights into this relationship in the context of multiple linear regression. We show, in particular, that the incorporation of a simple correlation measure into the tuning parameter can lead to a nearly optimal prediction performance of the Lasso even for highly correlated covariates. However, we also reveal that for moderately correlated covariates, the prediction performance of the Lasso can be mediocre irrespective of the choice of the tuning parameter. We finally show that our results also lead to near-optimal rates for the least-squares estimator with total variation penalty.

Popular sparse estimation methods based on $\ell_1$-relaxation, such as the Lasso and the Dantzig selector, require the knowledge of the variance of the noise in order to properly tune the regularization parameter. This constitutes a major obstacle in applying these methods in several frameworks---such as time series, random fields, inverse problems---for which the noise is rarely homoscedastic and its level is hard to know in advance. In this paper, we propose a new approach to the joint estimation of the conditional mean and the conditional variance in a high-dimensional (auto-) regression setting. An attractive feature of the proposed estimator is that it is efficiently computable even for very large scale problems by solving a second-order cone program (SOCP). We present theoretical analysis and numerical results assessing the performance of the proposed procedure.