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Debiased Estimators in High-Dimensional Regression: A Review and Replication of Javanmard and Montanari (2014)

Benjamin Smith

Abstract

High-dimensional statistical settings ($p \gg n$) pose fundamental challenges for classical inference, largely due to bias introduced by regularized estimators such as the LASSO. To address this, Javanmard and Montanari (2014) propose a debiased estimator that enables valid hypothesis testing and confidence interval construction. This report examines their debiased LASSO framework, which yields asymptotically normal estimators in high-dimensional settings. We present the key theoretical results underlying this approach, specifically, the construction of an optimized debiased estimator that restores asymptotic normality, which enables the computation of valid confidence intervals and $p$-values. To evaluate the claims of Javanmard and Montanari, a subset of the original simulation study and a re-examination of their real-data analysis are presented. Building on this baseline, we extend the empirical analysis to include the desparsified LASSO, a closely related method referenced but not implemented in the original study. The results demonstrate that while the debiased LASSO achieves reliable coverage and controls Type I error, the LASSO projection estimator can offer improved power in low-signal settings without compromising error rates. Our findings highlight a critical practical trade-off: while the LASSO projection estimator demonstrates superior statistical power in an idealized simulated low-signal setting, the estimation procedure employed by Javanmard and Montanari adapts more robustly to complex correlation networks, yielding superior precision and signal detection in real-world genomic data.

Debiased Estimators in High-Dimensional Regression: A Review and Replication of Javanmard and Montanari (2014)

Abstract

High-dimensional statistical settings () pose fundamental challenges for classical inference, largely due to bias introduced by regularized estimators such as the LASSO. To address this, Javanmard and Montanari (2014) propose a debiased estimator that enables valid hypothesis testing and confidence interval construction. This report examines their debiased LASSO framework, which yields asymptotically normal estimators in high-dimensional settings. We present the key theoretical results underlying this approach, specifically, the construction of an optimized debiased estimator that restores asymptotic normality, which enables the computation of valid confidence intervals and -values. To evaluate the claims of Javanmard and Montanari, a subset of the original simulation study and a re-examination of their real-data analysis are presented. Building on this baseline, we extend the empirical analysis to include the desparsified LASSO, a closely related method referenced but not implemented in the original study. The results demonstrate that while the debiased LASSO achieves reliable coverage and controls Type I error, the LASSO projection estimator can offer improved power in low-signal settings without compromising error rates. Our findings highlight a critical practical trade-off: while the LASSO projection estimator demonstrates superior statistical power in an idealized simulated low-signal setting, the estimation procedure employed by Javanmard and Montanari adapts more robustly to complex correlation networks, yielding superior precision and signal detection in real-world genomic data.

Paper Structure

This paper contains 13 sections, 5 theorems, 43 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Prerequisites
  4. A Debiased Estimator for $\theta_0$
  5. Statistical Inference
  6. Confidence Intervals
  7. Hypothesis Testing
  8. Generalizing to Simultaneous Confidence Intervals
  9. Non-Gaussian Noise
  10. Empirical Analysis
  11. Simulation Study
  12. Data Analysis
  13. Discussion & Conclusion

Key Result

Theorem 6

Let $\mathbf{X} \in \mathbb{R}^{n\times p}$ be any (determinisitic) design matrix and $\hat{\theta}^* = \hat{\theta}^*(Y, \mathbf{X}; M,\lambda)$ be the generalized debiased estimator as per Equation (eq:gen_debiased_theta). $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: A visualization of the symmetric circulant matrix $\Sigma$ specified mathematically in Equation (\ref{['eq:circulant_mat']})
  • Figure 2: Comparative high-dimensional inference on the riboflavin dataset ($n=71, p=4,088$). The Manhattan plots (left) display the global p-value distribution relative to a Bonferroni-corrected threshold (red line), while faceted forest plots (right) provide 95% confidence intervals for the top 10 genes.

Theorems & Definitions (14)

  • Definition 1: True Support Set of $\theta_0$
  • Definition 2: Sub-Gaussian Norm
  • Definition 3: Compatibility Constant
  • Definition 4: Generalized Coherence Parameter
  • Remark 5
  • Theorem 6: Error Decomposition around $\hat{\theta^*}$
  • proof
  • Theorem 7: Technicality
  • Theorem 8: Asymptotic Normality and Error Bounds for the Debiased LASSO Estimator $\hat{\theta}^u$
  • Remark 9
  • ...and 4 more