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Sparse Group Lasso: Optimal Sample Complexity, Convergence Rate, and Statistical Inference.

T Tony Cai1, Anru R Zhang2,3, Yuchen Zhou1,3

  • 1Department of Statistics & Data Science, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104.

IEEE Transactions on Information Theory
|March 3, 2023
PubMed
Summary
This summary is machine-generated.

We introduce sparse group Lasso for high-dimensional regression problems with double sparsity. This method achieves optimal sample complexity and estimation error bounds in both noiseless and noisy scenarios.

Keywords:
approximate dual certificateconvex optimizationsimultaneously structured modelsparse group Lassosparsity

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Area of Science:

  • Statistics
  • Machine Learning
  • High-Dimensional Data Analysis

Background:

  • Simultaneously structured models are crucial in statistics and machine learning.
  • Double sparsity (element-wise and group-wise) presents unique analytical challenges.
  • Existing methods may not fully address the complexities of double sparsity.

Purpose of the Study:

  • To develop and analyze sparse group Lasso for high-dimensional double sparse linear regression.
  • To establish theoretical bounds for sample complexity and estimation error.
  • To investigate the statistical inference capabilities of a debiased version of the method.

Main Methods:

  • Application of sparse group Lasso to a double sparse linear regression model.
  • Derivation of upper and lower bounds for sample complexity in noiseless settings.
  • Analysis of estimation error bounds and minimax lower bounds in noisy settings.
  • Investigation of asymptotic properties for debiased sparse group Lasso.

Main Results:

  • Matching upper and lower bounds on sample complexity for exact and stable recovery.
  • Obtained upper and matching minimax lower bounds for estimation error in the noisy case.
  • Established asymptotic properties of the debiased sparse group Lasso for inference.
  • Numerical studies validated the theoretical findings.

Conclusions:

  • Sparse group Lasso provides optimal performance for high-dimensional double sparse regression.
  • The theoretical results offer guarantees for both noiseless and noisy data scenarios.
  • The debiased method facilitates reliable statistical inference in these complex models.