Consistency of the group Lasso and multiple kernel learning

TL;DR

This paper analyzes the asymptotic consistency of the group Lasso and multiple kernel learning, providing necessary and sufficient conditions, and proposing an adaptive scheme for model estimation in high-dimensional and kernel-based settings.

Contribution

It extends the consistency analysis of the group Lasso to the infinite-dimensional kernel setting and introduces an adaptive method for consistent model estimation.

Findings

Derived necessary and sufficient conditions for group Lasso consistency.

Extended consistency results to multiple kernel learning with reproducing kernel Hilbert spaces.

Proposed an adaptive scheme that ensures consistency even under model misspecification.

Abstract

We consider the least-square regression problem with regularization by a block 1-norm, i.e., a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the 1-norm where all spaces have dimension one, where it is commonly referred to as the Lasso. In this paper, we study the asymptotic model consistency of the group Lasso. We derive necessary and sufficient conditions for the consistency of group Lasso under practical assumptions, such as model misspecification. When the linear predictors and Euclidean norms are replaced by functions and reproducing kernel Hilbert norms, the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection. Using tools from functional analysis, and in particular covariance operators, we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate, even when the necessary condition required for the non adaptive scheme is not satisfied.