Modeling with Categorical Features via Exact Fusion and Sparsity Regularisation

TL;DR

This paper introduces a novel high-dimensional linear regression method for categorical features that combines exact model compression with sparsity regularization, using mixed integer programming and efficient algorithms.

Contribution

It develops a new estimation approach with exact formulations and algorithms for clustering and sparsity, along with theoretical guarantees and improved performance.

Findings

Outperforms state-of-the-art methods on synthetic and real datasets.

Provides theoretical guarantees for prediction and cluster recovery.

Develops a fast approximate algorithm with high-quality solutions.

Abstract

We study the high-dimensional linear regression problem with categorical predictors that have many levels. We propose a new estimation approach, which performs model compression via two mechanisms by simultaneously encouraging (a) clustering of the regression coefficients to collapse some of the categorical levels together; and (b) sparsity of the regression coefficients. We present novel mixed integer programming formulations for our estimator, and develop a custom row generation procedure to speed up the exact off-the-shelf solvers. We also propose a fast approximate algorithm for our method that obtains high-quality feasible solutions via block coordinate descent. As the main building block of our algorithm, we develop an exact algorithm for the univariate case based on dynamic programming, which can be of independent interest. We establish new theoretical guarantees for both the prediction and the cluster recovery performance of our estimator. Our numerical experiments on synthetic and real datasets demonstrate that our proposed estimator tends to outperform the state-of-the-art.