---
abstract: |
    We present a detailed analysis of the class of regression decision tree algorithms which employ a regulized piecewise-linear node-splitting criterion and have regularized linear models at the leaves. From a theoretic standpoint, based on Rademacher complexity framework, we present new high-probability upper bounds for the generalization error for the proposed classes of regularized regression decision tree algorithms, including LASSO-type, and $\ell_{2}$ regularization for linear models at the leaves. Theoretical result are further extended by considering a general type of variable selection procedure. Furthermore, in our work we demonstrate that the analyzed class of regression trees is not only numerically stable but can furthermore be made tractable via an algorithmic implementation, presented herein, as well as with the help of modern GPU technology. Empirically, we present results on multiple datasets which highlight the strengths and potential pitfalls, of the proposed tree algorithms compared to baselines which grow trees based on piecewise constant models.

    **Keywords:** Decision trees, Piecewise-Linear Regression, Rademacher complexity, regularization.
author:
- 'Leonidas Lefakis, Oleksandr Zadorozhnyi, Gilles Blanchard'
bibliography:
- 'lrt.bib'
title: 'Efficient Regularized Piecewise-Linear Regression Trees '
---

Introduction {#sec:intro}
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Decision trees and random forests remain to be very popular machine learning tools because of 