Boosting for high-dimensional linear models

Early stopping for $ L^2 $-boosting in high-dimensional linear models

Increasingly high-dimensional data sets require that estimation methods do not only satisfy statistical guarantees but also remain computationally feasible. In this context, we consider $ L^{2} $-boosting via orthogonal matching pursuit in a high-dimensional linear model and analyze a data-driven early stopping time $ τ$ of the algorithm, which is sequential in the sense that its computation is based on the first $ τ$ iterations only. This approach is much less costly than established model selection criteria, that require the computation of the full boosting path. We prove that sequential ear…

Behavior of linear L2-boosting algorithms in the vanishing learning rate asymptotic

We investigate the asymptotic behaviour of gradient boosting algorithms when the learning rate converges to zero and the number of iterations is rescaled accordingly. We mostly consider L2-boosting for regression with linear base learner as studied in B{ü}hlmann and Yu (2003) and analyze also a stochastic version of the model where subsampling is used at each step (Friedman 2002). We prove a deterministic limit in the vanishing learning rate asymptotic and characterize the limit as the unique solution of a linear differential equation in an infinite dimensional function space. Besides, the tra…

On the Differences between L2-Boosting and the Lasso

We prove that L2-Boosting lacks a theoretical property which is central to the behaviour of l1-penalized methods such as basis pursuit and the Lasso: Whereas l1-penalized methods are guaranteed to recover the sparse parameter vector in a high-dimensional linear model under an appropriate restricted nullspace property, L2-Boosting is not guaranteed to do so. Hence, L2-Boosting behaves quite differently from l1-penalized methods when it comes to parameter recovery/estimation in high-dimensional linear models.

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

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 ext…

High-Dimensional $L_2$Boosting: Rate of Convergence

Boosting is one of the most significant developments in machine learning. This paper studies the rate of convergence of $L_2$Boosting, which is tailored for regression, in a high-dimensional setting. Moreover, we introduce so-called \textquotedblleft post-Boosting\textquotedblright. This is a post-selection estimator which applies ordinary least squares to the variables selected in the first stage by $L_2$Boosting. Another variant is \textquotedblleft Orthogonal Boosting\textquotedblright\ where after each step an orthogonal projection is conducted. We show that both post-$L_2$Boosting and the…

用于分组数据部分线性模型精确估计的Sandwich Boosting方法

Sandwich Boosting for Accurate Estimation in Partially Linear Models for Grouped Data

针对分组数据部分线性模型的参数估计问题，提出三明治损失与三明治提升方法，在协方差模型误设时可提升估计精度，并通过模拟和真实数据验证了有效性。