L1L_1-Penalization in Functional Linear Regression with Subgaussian Design

We study functional regression with random subgaussian design and real-valued response. The focus is on the problems in which the regression function can be well approximated by a functional linear model with the slope function being "sparse" in the sense that it can be represented as a sum of a small number of well separated "spikes". This can be viewed as an extension of now classical sparse estimation problems to the case of infinite dictionaries. We study an estimator of the regression function based on penalized empirical risk minimization with quadratic loss and the complexity penalty defined in terms of $L_1$-norm (a continuous version of LASSO). The main goal is to introduce several important parameters characterizing sparsity in this class of problems and to prove sharp oracle inequalities showing how the $L_2$-error of the continuous LASSO estimator depends on the underlying sparsity of the problem

Efficient median of means estimator

The goal of this note is to present a modification of the popular median of means estimator that achieves sub-Gaussian deviation bounds with nearly optimal constants under minimal assumptions on the underlying distribution. We build on a recent work on the topic by the author, and prove that desired guarantees can be attained under weaker requirements

Asymptotic normality of robust risk minimizers

This paper investigates asymptotic properties of a class of algorithms that can be viewed as robust analogues of the classical empirical risk minimization. These strategies are based on replacing the usual empirical average by a robust proxy of the mean, such as the (version of) the median-of-means estimator. It is well known by now that the excess risk of resulting estimators often converges to 0 at optimal rates under much weaker assumptions than those required by their "classical" counterparts. However, much less is known about the asymptotic properties of the estimators themselves, for instance, whether robust analogues of the maximum likelihood estimators are asymptotically efficient. We make a step towards answering these questions and show that for a wide class of parametric problems, minimizers of the appropriately defined robust proxy of the risk converge to the minimizers of the true risk at the same rate, and often have the same asymptotic variance, as the estimators obtained by minimizing the usual empirical risk. Moreover, our results show that robust algorithms based on the so-called "min-max" type procedures in many cases provably outperform, is the asymptotic sense, algorithms based on direct risk minimization