# A New Look at an Old Problem: A Universal Learning Approach to Linear Regression

arXiv: https://arxiv.org/abs/1905.04708 · HTML: https://mlsanity.com/paper/1905.04708/

## Key facts

- Cited by 11 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 12 May 2019.

## Built on by (most cited first, 11 of 11)

- [Fine-Tuning can Distort Pretrained Features and Underperform Out-of-Distribution](https://mlsanity.com/paper/2202.10054/) (2022)
- [Single Layer Predictive Normalized Maximum Likelihood for Out-of-Distribution Detection](https://mlsanity.com/paper/2110.09246/) (2021)
- [Generalization for multiclass classification with overparameterized linear models](https://mlsanity.com/paper/2206.01399/) (2022)
- [Bias-variance decomposition of overparameterized regression with random linear features](https://mlsanity.com/paper/2203.05443/) (2022)
- [The Geometry of Over-parameterized Regression and Adversarial Perturbations](https://mlsanity.com/paper/2103.14108/) (2021)
- [Mitigating deep double descent by concatenating inputs](https://mlsanity.com/paper/2107.00797/) (2021)
- [Distribution Free Uncertainty for the Minimum Norm Solution of Over-parameterized Linear Regression](https://mlsanity.com/paper/2102.07181/) (2021)
- [Utilizing Adversarial Targeted Attacks to Boost Adversarial Robustness](https://mlsanity.com/paper/2109.01945/) (2021)
- [Double Descent Optimization Pattern and Aliasing: Caveats of Noisy Labels](https://mlsanity.com/paper/2106.02100/) (2021)
- [Beyond Ridge Regression for Distribution-Free Data](https://mlsanity.com/paper/2206.08757/) (2022)
- [High Dimensional Binary Classification under Label Shift: Phase Transition and Regularization](https://mlsanity.com/paper/2212.00700/) (2022)

## Abstract (arXiv)

Linear regression is a classical paradigm in statistics. A new look at it is provided via the lens of universal learning. In applying universal learning to linear regression the hypotheses class represents the label y∈ cal R as a linear combination of the feature vector xTθ where x∈ cal RM, within a Gaussian error. The Predictive Normalized Maximum Likelihood (pNML) solution for universal learning of individual data can be expressed analytically in this case, as well as its associated learnability measure. Interestingly, the situation where the number of parameters M may even be larger than the number of training samples N can be examined. As expected, in this case learnability cannot be attained in every situation; nevertheless, if the test vector resides mostly in a subspace spanned by the eigenvectors associated with the large eigenvalues of the empirical correlation matrix of the training data, linear regression can generalize despite the fact that it uses an ``over-parametrized'' model. We demonstrate the results with a simulation of fitting a polynomial to data with a possibly large polynomial degree.