Variations of Lasso

Junwei Lu
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Junwei Lu: Harvard University

Chapter Chapter 11 in Big Data Analysis, 2025, pp 65-76 from Springer

Abstract: Abstract In the previous chapter, we study the high-dimensional linear model Y = 𝕏 β ∗ + 𝜖 $$Y = \mathbb {X}\beta ^* + \epsilon $$ , with 𝕏 ∈ ℝ n × d $$\mathbb {X} \in \mathbb {R}^{n \times d}$$ and ∥ β ∗ ∥ 0 ≤ s $$\|\beta ^*\|_0\le s$$ . We propose to estimate β ∗ $$\beta ^*$$ via Lasso estimator β ^ Lasso = arg min β 1 2 n ∥ Y − 𝕏 β ∥ 2 2 + λ ∥ β ∥ 1 . $$\displaystyle \widehat \beta ^{\mathrm {Lasso}} = \operatorname *{\text{arg min}}_{\beta} \frac {1}{2n}\|Y - \mathbb {X} \beta \|_2^2 + \lambda \|\beta \|_1. $$ We consider two assumptions: (1) the design matrix satisfies the restricted eigenvalue condition and (2) the noises ε $$\varepsilon $$ are independent sub-Gaussians with variance proxy σ 2 $$\sigma ^2$$ . If we choose λ = Cσ log d ∕ n $$\lambda = C\sigma \sqrt {\log d/n}$$ for some sufficiently large constant C, we show that the Lasso estimator has the statistical rate ∥ β ^ Lasso − β ∗ ∥ 2 = O P ( s log d ∕ n ) $$\| \widehat \beta ^{\mathrm {Lasso}} - \beta ^*\|_2 = O_P(\sqrt {s\log d/n})$$ .

Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-03161-7_11

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DOI: 10.1007/978-3-032-03161-7_11

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