Localized Lasso for High-Dimensional Regression

TL;DR

The paper presents the localized Lasso, a convex, interpretable high-dimensional regression method that uses local sparse models with network regularization and exclusive group sparsity, optimized efficiently without tuning parameters.

Contribution

It introduces the localized Lasso with a novel combination of regularizations and an efficient optimization algorithm, improving interpretability and predictive power in high-dimensional, small-sample settings.

Findings

Outperforms alternative methods on simulated data

Effective in genomic personalized medicine datasets

Convex cost function guarantees global optimality

Abstract

We introduce the localized Lasso, which is suited for learning models that are both interpretable and have a high predictive power in problems with high dimensionality $d$ and small sample size $n$. More specifically, we consider a function defined by local sparse models, one at each data point. We introduce sample-wise network regularization to borrow strength across the models, and sample-wise exclusive group sparsity (a.k.a., $\ell_{1,2}$ norm) to introduce diversity into the choice of feature sets in the local models. The local models are interpretable in terms of similarity of their sparsity patterns. The cost function is convex, and thus has a globally optimal solution. Moreover, we propose a simple yet efficient iterative least-squares based optimization procedure for the localized Lasso, which does not need a tuning parameter, and is guaranteed to converge to a globally optimal solution. The solution is empirically shown to outperform alternatives for both simulated and genomic personalized medicine data.