Fast Debiasing of the LASSO Estimator (2502.19825v1)

Abstract: In high-dimensional sparse regression, the \textsc{Lasso} estimator offers excellent theoretical guarantees but is well-known to produce biased estimates. To address this, \cite{Javanmard2014} introduced a method to debias" the \textsc{Lasso} estimates for a random sub-Gaussian sensing matrix $\boldsymbol{A}$. Their approach relies on computing anapproximate inverse" $\boldsymbol{M}$ of the matrix $\boldsymbol{A}^\top \boldsymbol{A}/n$ by solving a convex optimization problem. This matrix $\boldsymbol{M}$ plays a critical role in mitigating bias and allowing for construction of confidence intervals using the debiased \textsc{Lasso} estimates. However the computation of $\boldsymbol{M}$ is expensive in practice as it requires iterative optimization. In the presented work, we re-parameterize the optimization problem to compute a ``debiasing matrix" $\boldsymbol{W} := \boldsymbol{AM}^{\top}$ directly, rather than the approximate inverse $\boldsymbol{M}$. This reformulation retains the theoretical guarantees of the debiased \textsc{Lasso} estimates, as they depend on the \emph{product} $\boldsymbol{AM}^{\top}$ rather than on $\boldsymbol{M}$ alone. Notably, we provide a simple, computationally efficient, closed-form solution for $\boldsymbol{W}$ under similar conditions for the sensing matrix $\boldsymbol{A}$ used in the original debiasing formulation, with an additional condition that the elements of every row of $\boldsymbol{A}$ have uncorrelated entries. Also, the optimization problem based on $\boldsymbol{W}$ guarantees a unique optimal solution, unlike the original formulation based on $\boldsymbol{M}$. We verify our main result with numerical simulations.

• The paper introduces a computationally efficient method for debiasing the Lasso estimator in high-dimensional sparse regression, addressing bias caused by regularization.
• It proposes reparameterizing the optimization problem and provides a closed-form solution for the debiasing matrix \(\boldsymbol{W}\) under specific conditions on the sensing matrix, significantly reducing computational cost.
• The new method achieves equivalent theoretical guarantees and performance as prior approaches but enables much faster computation, facilitating real-time applications and hypothesis testing.

Fast Debiasing of the Lasso Estimator: A Computationally Efficient Approach

The paper "Fast Debiasing of the LASSO Estimator" by Shuvayan Banerjee et al. addresses a significant challenge in high-dimensional sparse regression. Specifically, it focuses on the bias inherent in Lasso estimators—widely used for variable selection and estimation—owing to the penalty regularization that promotes sparsity. Despite the Lasso's potent theoretical guarantees, its bias can adversely impact the quality of parameter estimates and the validity of statistical inferences such as hypothesis tests.

The seminal work by Javanmard and Montanari (2014) introduced a method for "debiasing" the Lasso estimates, thus overcoming some of the associated limitations. This approach involves calculating an approximate inverse of the sample covariance matrix, $\boldsymbol{M}$, by solving a convex optimization problem. This inverse is employed to correct for bias and facilitate the construction of valid confidence intervals under certain random sensing conditions.

In this paper, the authors propose a novel reformulation of the optimization problem by introducing a "debiasing matrix" $\boldsymbol{W} = \boldsymbol{A}\boldsymbol{M}^\top$, simplifying the debiasing process while maintaining the theoretical properties of the original approach. Notably, they provide a computationally efficient, closed-form solution for $\boldsymbol{W}$ when the rows of the sensing matrix $\boldsymbol{A}$ contain uncorrelated entries.

Key Contributions

1. Reparameterization of the Optimization Problem: By focusing on the product $$\boldsymbol{W} := \boldsymbol{A}\boldsymbol{M}^\top$$ rather than $\boldsymbol{M}$ directly, the optimization task is substantially reduced. This reformulation is pivotal because the properties of the debiased Lasso solely depend on $\boldsymbol{W}$, allowing the authors to sidestep the computationally intensive iterative optimization that $\boldsymbol{M}$ would require.
2. Closed-form Solution: Under the assumption of sensing matrices with independent, identically distributed sub-Gaussian rows, the authors derive a closed-form solution for $\boldsymbol{W}$. This solution also assumes that the elements of $\boldsymbol{A}$ are uncorrelated, a condition satisfied with high probability in many applications involving isotropic random matrices.
3. Theoretical Guarantees: The paper rigorously establishes that the debiased estimator retains asymptotic normality, enabling effective hypothesis testing in high-dimensional regimes. This ensures continued applicability in scenarios where $\boldsymbol{A}$ adheres to particular probabilistic structures.

Numerical Simulations and Implications

The authors verify their theoretical results with simulations, demonstrating that the closed-form solution for $\boldsymbol{W}$ provides results that are equivalent to those obtained via the original method, while offering significant computational savings. This advancement empowers practitioners to deploy the debiasing process in real-time scenarios or applications with stringent computational constraints.

Future Directions

This advancement in debiasing methodology could inspire further research into robust and efficient implementations for other types of regularization problems. As computational efficiency is enhanced, the applicability of high-dimensional statistical methods in machine learning and AI could greatly expand, particularly in fields requiring rapid, real-time inference such as genomics or real-time image processing.

In conclusion, "Fast Debiasing of the LASSO Estimator" contributes a valuable methodological refinement to the area of high-dimensional statistics, offering both a theoretical framework and practical tools for efficient sparse regression in complex datasets.