# Optimal Bias-Correction and Valid Inference in High-Dimensional Ridge Regression: A Closed-Form Solution

arXiv: https://arxiv.org/abs/2405.00424 · HTML: https://mlsanity.com/paper/2405.00424/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 31 references that are in the map.
- First posted on arXiv 1 May 2024.

## Builds on (most cited first, 6 of 6)

- [High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification](https://mlsanity.com/paper/1507.03003/) (2015)
- [Machine Learning Methods Economists Should Know About](https://mlsanity.com/paper/1903.10075/) (2019)
- [Lecture notes on ridge regression](https://mlsanity.com/paper/1509.09169/) (2015)
- [High-dimensional Ordinary Least-squares Projection for Screening Variables](https://mlsanity.com/paper/1506.01782/) (2015)
- [Ridge Regression Revisited: Debiasing, Thresholding and Bootstrap](https://mlsanity.com/paper/2009.08071/) (2020)
- [Supervised Dynamic PCA: Linear Dynamic Forecasting with Many Predictors](https://mlsanity.com/paper/2307.07689/) (2023)

## Abstract (arXiv)

Ridge regression is an indispensable tool in big data analysis. Yet its inherent bias poses a significant and longstanding challenge, compromising both statistical efficiency and scalability across various applications. To tackle this critical issue, we introduce an iterative strategy to correct bias effectively when the dimension p is less than the sample size n. For p>n, our method optimally mitigates the bias such that any remaining bias in the proposed de-biased estimator is unattainable through linear transformations of the response data. To address the remaining bias when p>n, we employ a Ridge-Screening (RS) method, producing a reduced model suitable for bias correction. Crucially, under certain conditions, the true model is nested within our selected one, highlighting RS as a novel variable selection approach. Through rigorous analysis, we establish the asymptotic properties and valid inferences of our de-biased ridge estimators for both p<n and p>n, where, both p and n may increase towards infinity, along with the number of iterations. We further validate these results using simulated and real-world data examples. Our method offers a transformative solution to the bias challenge in ridge regression inferences across various disciplines.