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

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Variations of Lasso

  • 章节

  • 发表日期：

    2025年

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  • Junwei Lu

摘要

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