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

By Koby Bibas, Yaniv Fogel, Meir Feder. 2019

36 citations recorded by the source corpus.

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 ∈ ℛ as a linear combination of the feature vector xT θ where x ∈ 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.

Subfields: Artificial Intelligence, Computational Mechanics, Computer Vision and Pattern Recognition, Statistics and Probability, Statistical Learning Theory

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Benoit Sanchez
2. Jonathan Lomond
3. Dmitry Kobak

In Computational Mechanics

1. Benoit Sanchez
2. Jonathan Lomond
3. Dmitry Kobak

In Computer Vision and Pattern Recognition

1. Benoit Sanchez
2. Jonathan Lomond
3. Dmitry Kobak

In Statistics and Probability

1. Benoit Sanchez
2. Jonathan Lomond
3. Dmitry Kobak

In Statistical Learning Theory

1. Benoit Sanchez
2. Jonathan Lomond
3. Dmitry Kobak

Citations in the corpus

• Deep double descent: where bigger models and more data hurt (2019, neutral): “Our notion of Effective Model Complexity is related to classical complexity notions such as Rademacher complexity, but differs in several crucial ways: (1) EMC depends on the true labels of the data distribution, and (2) EMC depends on the training procedure, not just the model architecture.”
• The Optimal Ridge Penalty for Real-world High-dimensional Data Can Be Zero or Negative due to the Implicit Ridge Regularization (2020, background): “Finally, finding (iv) is related to the results of Bibas et al. (2019) and Bartlett et al. (2020); the connection between the minimum-norm OLS and the ridge estimators was also studied by Dereziński et al. (2019).”
• Fine-Tuning can Distort Pretrained Features and Underperform Out-of-Distribution (2022, unknown): “…analyzing different linear overparameterized settings Belkin et al. (2019); Hastie et al. (2019); Bartlett et al. (2019); Muthukumar et al. (2020); Mei & Montanari (2019); Bibas et al. (2019) study various statistical properties such as the “double descent curve” in addition to benign overfitting.”