# A Simple Approximation to the Distribution of the Ridge Regression Estimator

arXiv: https://arxiv.org/abs/2608.02539 · HTML: https://mlsanity.com/paper/2608.02539/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 35 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Builds on (most cited first, 7 of 7)

- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification](https://mlsanity.com/paper/1507.03003/) (2015)
- [Exact minimax risk for linear least squares, and the lower tail of sample covariance matrices](https://mlsanity.com/paper/1912.10754/) (2019)
- [Optimal Ridge Regularization for Out-of-Distribution Prediction](https://mlsanity.com/paper/2404.01233/) (2024)
- [An elementary analysis of ridge regression with random design](https://mlsanity.com/paper/2203.08564/) (2022)
- [Risk and cross validation in ridge regression with correlated samples](https://mlsanity.com/paper/2408.04607/) (2024)
- [Inference in Linear Regression Models with Many Covariates and Heteroskedasticity](https://mlsanity.com/paper/1507.02493/) (2015)

## Abstract (arXiv)

We present a simple Gaussian approximation to the finite-sample distribution of the classical ridge regression estimator. Our approximation captures the fact that, in finite samples, the ridge regression estimator trades off bias and variance to reduce estimation and prediction error. Our approximation is based on nonstandard asymptotics where i) we let the estimator's regularization parameter grow proportionally to the sample size; and ii) we treat the population regression coefficients as local to the reference vector that defines the estimator's direction of shrinkage. In contrast to other asymptotic approximations in the literature, we allow for general forms of heteroskedasticity and autocorrelation in the data generating process (at the cost of considering a low-dimensional model where the number of covariates is not allowed to grow with the sample size). We use our simple Gaussian approximation to propose two new strategies to select the regularization parameter for the ridge regression estimator. The suggested strategies select the regularization parameter to minimize either average or worst-case excess prediction risk, where risk is computed using our suggested Gaussian approximation.