Sharp Oracle Inequalities for Square Root Regularization
Authors: Benjamin Stucky, Sara van de Geer
JMLR 2017 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Based on a simulation we illustrate some advantages of the square root SLOPE. (Abstract) ... The goal of this simulation is to see how the estimation and prediction errors for the square root LASSO and the square root SLOPE behave under some Gaussian designs. ... The results can be found in Table 1,2,3 and 4. |
| Researcher Affiliation | Academia | Benjamin Stucky EMAIL Seminar for Statistics ETH Z urich R amistrasse 101 8092 Zurich, Switzerland Sara van de Geer EMAIL Seminar for Statistics ETH Z urich R amistrasse 101 8092 Zurich, Switzerland |
| Pseudocode | Yes | Algorithm 1: sr SLOPE input : β0 a starting parameter vector, λ a desired penalty level with a decreasing sequence, Y the response vector, X the design matrix. output: ˆβsr SLOPE = arg min β Rp ( Y Xβ n + λJλ(β)) 1 for i 0 to istop do 2 σi+1 Y Xβi n; 3 βi+1 arg min β Rp Y Xβ 2 n + σi+1λJλ(β) ; |
| Open Source Code | No | For the square root LASSO we have used the R-Package flare by Li et al. (2014). |
| Open Datasets | No | The design matrix X is chosen with the rows being fixed i.i.d. realizations from N(0, Σ). Here the covariance matrix Σ has a Toeplitz structure Σi,j = 0.9|i j|. We choose i.i.d. Gaussian errors ϵ with a variance of σ2 = 1. |
| Dataset Splits | No | We consider a high-dimensional linear regression model: Y = Xβ0 + ϵ, with n = 100 response variables and p = 500 unknown parameters. ... We use r = 100 repetitions to calculate the ℓ1 estimation error, the sorted ℓ1 estimation error and the ℓ2 prediction error. |
| Hardware Specification | No | The paper does not provide specific hardware details for running the experiments. |
| Software Dependencies | Yes | For the square root LASSO we have used the R-Package flare by Li et al. (2014). |
| Experiment Setup | Yes | We consider a high-dimensional linear regression model: Y = Xβ0 + ϵ, with n = 100 response variables and p = 500 unknown parameters. The design matrix X is chosen with the rows being fixed i.i.d. realizations from N(0, Σ). Here the covariance matrix Σ has a Toeplitz structure Σi,j = 0.9|i j|. We choose i.i.d. Gaussian errors ϵ with a variance of σ2 = 1. ... As for the definition of the sorted ℓ1 norm, we chose a regular decreasing sequence from 1 to 0.1 with length 500. We use r = 100 repetitions... |