Adaptive Lasso, Transfer Lasso, and Beyond: An Asymptotic Perspective (2308.15838v2)

Abstract: This paper presents a comprehensive exploration of the theoretical properties inherent in the Adaptive Lasso and the Transfer Lasso. The Adaptive Lasso, a well-established method, employs regularization divided by initial estimators and is characterized by asymptotic normality and variable selection consistency. In contrast, the recently proposed Transfer Lasso employs regularization subtracted by initial estimators with the demonstrated capacity to curtail non-asymptotic estimation errors. A pivotal question thus emerges: Given the distinct ways the Adaptive Lasso and the Transfer Lasso employ initial estimators, what benefits or drawbacks does this disparity confer upon each method? This paper conducts a theoretical examination of the asymptotic properties of the Transfer Lasso, thereby elucidating its differentiation from the Adaptive Lasso. Informed by the findings of this analysis, we introduce a novel method, one that amalgamates the strengths and compensates for the weaknesses of both methods. The paper concludes with validations of our theory and comparisons of the methods via simulation experiments.

• The paper introduces a theoretical comparison of Adaptive and Transfer Lasso methods, revealing that the latter generally lacks the oracle property.
• It employs rigorous simulations to map parameter phase diagrams, highlighting regions of √n- and √m-consistency along with variable selection success.
• The study presents the new Adaptive Transfer Lasso method, which synergizes dual regularization to enhance asymptotic convergence and practical estimator performance.

An Asymptotic Analysis of Adaptive, Transfer, and Integrated Lasso Methods

This paper offers an in-depth exploration of the theoretical characteristics of the Adaptive Lasso and the Transfer Lasso and goes beyond by introducing a new method, the Adaptive Transfer Lasso. While the Adaptive Lasso is established with its capacity to achieve variable selection consistency and asymptotic normality, the Transfer Lasso has been proposed for sparing non-asymptotic estimation errors, posing unique differences in its approach to employing initial estimators.

Core Contributions

1. Theoretical Foundation and Comparison:
   • The Adaptive Lasso uses an initial $\sqrt{n}$-consistent estimator and adjusts regularization weights to shrink some coefficients to zero. Its oracle property for achieving both $\sqrt{n}$-consistency and variable selection consistency under suitable conditions is analytically known.
   • The Transfer Lasso integrates knowledge from an initial estimator to guide the sparsification process, adding a regularization term involving the difference between the target and initial parameters.
   • The paper demonstrates theoretically that the Transfer Lasso, although advantageous in various convergence rates especially when initial estimators come from large datasets, does not generally achieve the oracle property.
2. Empirical and Asymptotic Investigations:
   • The authors present parameter phase diagrams through simulations to illustrate regions of $\sqrt{n}$-consistency, $\sqrt{m}$-consistency, and variable selection consistency for both Lasso methods.
   • For the Adaptive Lasso, regions where strong theoretical properties hold (such as combined consistency of estimation and selection) are further clarified with assumptions on the sample sizes of initial estimators.
3. The Adaptive Transfer Lasso:
   • As a novel contribution, the Adaptive Transfer Lasso synergizes strategies from both Adaptive and Transfer Lasso methods. It integrates dual regularization terms adjusting weights influenced by initial estimators, thus attempting to harmonize the favorable qualities of both pre-existing methods.
   • Through rigorous analysis, it establishes the conditions under which $\sqrt{m}$-consistency and strong variable selection properties jointly exist, extending the potential of Lasso-type estimators when leveraging cross-domain initial estimators.

Implications

This research advances the statistical foundations of high-dimensional regression estimators by formalizing how initial estimators influence asymptotic properties, especially in transfer learning contexts. Implications stretch from practical, large-scale regression tasks where accurate selection of active features is imperative, to more theoretical considerations on estimator behavior in asymptotic regimes. The introduction of a versatile Adaptive Transfer Lasso offers a promising tool for statisticians and data scientists alike, advancing the toolkit available for tackling complex sparsity and estimation problems.

Speculation on AI and Future Directions

The findings presented shape new possibilities for AI systems, particularly in domains requiring adaptive statistical models underpinned by strong asymptotic guarantees. As machine learning increasingly emphasizes transferability and scalability across varied tasks, the Adaptive Transfer Lasso could provide robust solutions in dynamically varying environments. Future research may delve into high-dimensional asymptotics where $p \gg n$, solidifying the theoretical underpinnings for even broader application scenarios.

This paper marks a stimulating intellectual contribution to the Lasso family of estimators, balancing between novel theoretical insights and empirical validation, with practical implications for adaptive and transfer learning methodologies.