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Scaled Sparse Linear Regression (1104.4595v2)

Published 24 Apr 2011 in stat.ML, math.ST, and stat.TH

Abstract: Scaled sparse linear regression jointly estimates the regression coefficients and noise level in a linear model. It chooses an equilibrium with a sparse regression method by iteratively estimating the noise level via the mean residual square and scaling the penalty in proportion to the estimated noise level. The iterative algorithm costs little beyond the computation of a path or grid of the sparse regression estimator for penalty levels above a proper threshold. For the scaled lasso, the algorithm is a gradient descent in a convex minimization of a penalized joint loss function for the regression coefficients and noise level. Under mild regularity conditions, we prove that the scaled lasso simultaneously yields an estimator for the noise level and an estimated coefficient vector satisfying certain oracle inequalities for prediction, the estimation of the noise level and the regression coefficients. These inequalities provide sufficient conditions for the consistency and asymptotic normality of the noise level estimator, including certain cases where the number of variables is of greater order than the sample size. Parallel results are provided for the least squares estimation after model selection by the scaled lasso. Numerical results demonstrate the superior performance of the proposed methods over an earlier proposal of joint convex minimization.

Citations (496)

Summary

  • The paper offers a novel scaled lasso approach that estimates both regression coefficients and noise levels simultaneously for improved accuracy.
  • It employs an iterative algorithm based on gradient descent within a convex minimization framework to dynamically adjust the penalty scaling.
  • Numerical simulations and theoretical analyses confirm enhanced prediction accuracy and robustness, even when variables outnumber observations.

Scaled Sparse Linear Regression: A Technical Overview

The paper "Scaled Sparse Linear Regression" by Tingni Sun and Cun-Hui Zhang presents an advanced approach to linear regression, particularly in high-dimensional settings where the number of variables exceeds the number of observations. This method integrates the estimation of regression coefficients and noise level, utilizing a scaled lasso algorithm, an evolution of the traditional lasso.

Core Concepts and Methodology

Scaled sparse linear regression addresses the challenges of determining appropriate penalty levels, which are critical in ensuring model simplicity while avoiding excessive bias. The approach proposed by Sun and Zhang iteratively adjusts the penalty according to the estimated noise, leading to an equilibrium state defined by the scaled lasso.

Key steps involve:

  • Iteratively estimating the noise level using the mean residual square.
  • Scaling the penalty proportionally to the noise level.

The method leverages a gradient descent strategy within a convex minimization framework, offering an efficient computational solution. The authors assert that under certain regularity conditions, this approach achieves consistency and asymptotic normality, even when the variable count surpasses the sample size.

Numerical Results and Theoretical Implications

Numerical simulations demonstrate that the scaled lasso provides superior performance compared to earlier methods, specifically in terms of prediction accuracy and noise level estimation. In certain test scenarios, the scaled lasso remains resilient against high-dimensional challenges, maintaining effectiveness without prior knowledge of the noise level.

Theoretical analysis unveils oracle inequalities which guarantee the robustness of the scaled lasso method concerning prediction and parameter estimation. These inequalities establish conditions that ensure the method's consistency, facilitating predictions and noise estimations that are asymptotically normal.

Implications for High-Dimensional Data Analysis

For practical applications, the scaled sparse regression method holds significant promise in fields requiring extensive variable analysis, such as genomics or finance, where variable selection and noise estimation complexities are prevalent. By achieving scale invariance, the approach mitigates common pitfalls associated with variable selection in noisy data environments.

Additionally, the method's efficiency reduces computational load, a crucial factor in practical applications involving massive datasets. The paper hints at further versatility by suggesting extensions to other penalized regression frameworks, propelled by the method's flexible penalty scaling mechanism.

Future Directions

The authors suggest that future research could optimize penalty levels, potentially adapting them through various statistical measures to refine prediction error minimization. The theoretical foundations laid out in this work open avenues for further exploration of concave penalties and adjustments for degrees of freedom, promising deeper insights into optimal model selection strategies.

In conclusion, this paper contributes significantly to the understanding and implementation of scalable high-dimensional regression techniques, providing a robust alternative to existing methodologies and setting a benchmark for future innovations in sparse regression models.