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A Note on Estimation Error Bound and Grouping Effect of Transfer Elastic Net

Yui Tomo

TL;DR

This work analyzes Transfer Elastic Net (TENet) for high-dimensional linear regression by deriving a non-asymptotic $\\ell_2$-norm estimation bound under sub-Gaussian errors and a generalized restricted eigenvalue condition. The core result provides an explicit bound $U_{TENet}$ that depends on regularization, transfer strength, and source-target mismatch, and shows that TENet can outperform both Elastic Net and Transfer Lasso under strong problem relatedness. The authors also prove a grouping effect for TENet, demonstrating that coefficients for highly correlated predictors converge when source estimates are close, with explicit bounds involving predictor correlations and transfer parameters. Additional results establish when the generalized RE condition holds for Gaussian designs and relate TENet bounds to those of ENet and TLasso, highlighting conditions under which transfer improves estimation accuracy and stability in the presence of highly correlated features.

Abstract

The Transfer Elastic Net is an estimation method for linear regression models that combines $\ell_1$ and $\ell_2$ norm penalties to facilitate knowledge transfer. In this study, we derive a non-asymptotic $\ell_2$ norm estimation error bound for the estimator and discuss scenarios where the Transfer Elastic Net effectively works. Furthermore, we examine situations where it exhibits the grouping effect, which states that the estimates corresponding to highly correlated predictors have a small difference.

A Note on Estimation Error Bound and Grouping Effect of Transfer Elastic Net

TL;DR

This work analyzes Transfer Elastic Net (TENet) for high-dimensional linear regression by deriving a non-asymptotic -norm estimation bound under sub-Gaussian errors and a generalized restricted eigenvalue condition. The core result provides an explicit bound that depends on regularization, transfer strength, and source-target mismatch, and shows that TENet can outperform both Elastic Net and Transfer Lasso under strong problem relatedness. The authors also prove a grouping effect for TENet, demonstrating that coefficients for highly correlated predictors converge when source estimates are close, with explicit bounds involving predictor correlations and transfer parameters. Additional results establish when the generalized RE condition holds for Gaussian designs and relate TENet bounds to those of ENet and TLasso, highlighting conditions under which transfer improves estimation accuracy and stability in the presence of highly correlated features.

Abstract

The Transfer Elastic Net is an estimation method for linear regression models that combines and norm penalties to facilitate knowledge transfer. In this study, we derive a non-asymptotic norm estimation error bound for the estimator and discuss scenarios where the Transfer Elastic Net effectively works. Furthermore, we examine situations where it exhibits the grouping effect, which states that the estimates corresponding to highly correlated predictors have a small difference.

Paper Structure

This paper contains 8 sections, 9 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Estimation Error Bound
  3. Grouping Effect
  4. Proof of Theorem \ref{['thm:estimation_error_tenet']}
  5. Proof of Proposition \ref{['prop:tenet_vs_enet']}
  6. Proof of Proposition \ref{['prop:tenet_vs_tlasso']}
  7. Proof of Proposition \ref{['prop:sample_X_GRE']}
  8. Proof of Theorem \ref{['thm:grouping_effect']}

Key Result

Theorem 1

Suppose that Assumption ast:subgaussian and Assumption ast:gre are satisfied. We define where $\phi_{\mathrm{TENet}} = \phi(\mathcal{B}(\alpha, \rho, c, \boldsymbol{\Delta}))$. Then we have with probability at least $1-\exp(-n c^2 \lambda^2 /2\sigma^2 + \log(2p))$.

Theorems & Definitions (18)

  • Theorem 1
  • Corollary 2
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 6
  • Remark 2
  • Lemma 7: Technical Lemma
  • proof
  • ...and 8 more