A Data-Driven Line Search Rule for Support Recovery in High-dimensional Data Analysis
This work addresses a practical bottleneck in high-dimensional data analysis for researchers and practitioners, but it is incremental as it builds on existing ideas like support detection and root finding.
The authors tackled the problem of selecting step sizes in ℓ0-penalized regression algorithms, which typically rely on fixed step sizes that are hard to compute due to restrictive assumptions, by proposing a data-driven line search rule. The result showed improved stability, effectiveness, and superiority in numerical comparisons on linear and logistic regression problems, though no concrete numbers were provided.
In this work, we consider the algorithm to the (nonlinear) regression problems with $\ell_0$ penalty. The existing algorithms for $\ell_0$ based optimization problem are often carried out with a fixed step size, and the selection of an appropriate step size depends on the restricted strong convexity and smoothness for the loss function, hence it is difficult to compute in practical calculation. In sprite of the ideas of support detection and root finding \cite{HJK2020}, we proposes a novel and efficient data-driven line search rule to adaptively determine the appropriate step size. We prove the $\ell_2$ error bound to the proposed algorithm without much restrictions for the cost functional. A large number of numerical comparisons with state-of-the-art algorithms in linear and logistic regression problems show the stability, effectiveness and superiority of the proposed algorithms.