A New Perspective on Boosting in Linear Regression via Subgradient Optimization and Relatives

TL;DR

This paper reinterprets classic boosting algorithms in linear regression through the lens of modern convex optimization, offering new algorithms, theoretical guarantees, and insights into their statistical properties and regularization effects.

Contribution

It introduces a novel perspective by linking boosting algorithms to subgradient methods, proposes modifications for Lasso, and provides comprehensive computational guarantees.

Findings

Boosting algorithms can be viewed as subgradient descent on a max correlation loss.

New algorithms for Lasso derived from boosting methods.

Theoretical guarantees on data-fidelity and regularization effects.

Abstract

In this paper we analyze boosting algorithms in linear regression from a new perspective: that of modern first-order methods in convex optimization. We show that classic boosting algorithms in linear regression, namely the incremental forward stagewise algorithm (FS$_\varepsilon$) and least squares boosting (LS-Boost($\varepsilon$)), can be viewed as subgradient descent to minimize the loss function defined as the maximum absolute correlation between the features and residuals. We also propose a modification of FS$_\varepsilon$ that yields an algorithm for the Lasso, and that may be easily extended to an algorithm that computes the Lasso path for different values of the regularization parameter. Furthermore, we show that these new algorithms for the Lasso may also be interpreted as the same master algorithm (subgradient descent), applied to a regularized version of the maximum absolute correlation loss function. We derive novel, comprehensive computational guarantees for several boosting algorithms in linear regression (including LS-Boost($\varepsilon$) and FS$_\varepsilon$) by using techniques of modern first-order methods in convex optimization. Our computational guarantees inform us about the statistical properties of boosting algorithms. In particular they provide, for the first time, a precise theoretical description of the amount of data-fidelity and regularization imparted by running a boosting algorithm with a prespecified learning rate for a fixed but arbitrary number of iterations, for any dataset.