Shrinkage Degree in $L_{2}$ -Rescale Boosting for Regression
<inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-rescale boosting (<inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-RBoosting) is a variant of <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-Boosting, which can essentially improve the generalization performance of <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-Boosting. The key feature of <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-RBoosting lies in introducing a shrinkage degree to rescale the ensemble estimate in each iteration. Thus, the shrinkage degree determines the performance of <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-RBoosting. The aim of this paper is to develop a concrete analysis concerning how to determine the shrinkage degree in <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-RBoosting. We propose two feasible ways to select the shrinkage degree. The first one is to parameterize the shrinkage degree and the other one is to develop a data-driven approach. After rigorously analyzing the importance of the shrinkage degree in <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-RBoosting, we compare the pros and cons of the proposed methods. We find that although these approaches can reach the same learning rates, the structure of the final estimator of the parameterized approach is better, which sometimes yields a better generalization capability when the number of sample is finite. With this, we recommend to parameterize the shrinkage degree of <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-RBoosting. We also present an adaptive parameter-selection strategy for shrinkage degree and verify its feasibility through both theoretical analysis and numerical verification. The obtained results enhance the understanding of <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-RBoosting and give guidance on how to use it for regression tasks.