# Interpolating Predictors in High-Dimensional Factor Regression

arXiv: https://arxiv.org/abs/2002.02525 · HTML: https://mlsanity.com/paper/2002.02525/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Feb 2020.

## Built on by (most cited first, 3 of 3)

- [Towards an Understanding of Benign Overfitting in Neural Networks](https://mlsanity.com/paper/2106.03212/) (2021)
- [The Interplay Between Implicit Bias and Benign Overfitting in Two-Layer Linear Networks](https://mlsanity.com/paper/2108.11489/) (2021)
- [Deep Linear Networks can Benignly Overfit when Shallow Ones Do](https://mlsanity.com/paper/2209.09315/) (2022)

## Abstract (arXiv)

This work studies finite-sample properties of the risk of the minimum-norm interpolating predictor in high-dimensional regression models. If the effective rank of the covariance matrix Sigma of the p regression features is much larger than the sample size n, we show that the min-norm interpolating predictor is not desirable, as its risk approaches the risk of trivially predicting the response by 0. However, our detailed finite-sample analysis reveals, surprisingly, that this behavior is not present when the regression response and the features are it jointly low-dimensional, following a widely used factor regression model. Within this popular model class, and when the effective rank of Sigma is smaller than n, while still allowing for p gg n, both the bias and the variance terms of the excess risk can be controlled, and the risk of the minimum-norm interpolating predictor approaches optimal benchmarks. Moreover, through a detailed analysis of the bias term, we exhibit model classes under which our upper bound on the excess risk approaches zero, while the corresponding upper bound in the recent work arXiv:1906.11300 diverges. Furthermore, we show that the minimum-norm interpolating predictor analyzed under the factor regression model, despite being model-agnostic and devoid of tuning parameters, can have similar risk to predictors based on principal components regression and ridge regression, and can improve over LASSO based predictors, in the high-dimensional regime.