TY - GEN

T1 - A New Look at an Old Problem

T2 - 2019 IEEE International Symposium on Information Theory, ISIT 2019

AU - Bibas, Koby

AU - Fogel, Yaniv

AU - Feder, Meir

N1 - Publisher Copyright:
© 2019 IEEE.

PY - 2019/7

Y1 - 2019/7

N2 - 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 ∈ as a linear combination of the feature vector xT θ where x ∈ M, 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.

AB - 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 ∈ as a linear combination of the feature vector xT θ where x ∈ M, 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.

UR - http://www.scopus.com/inward/record.url?scp=85073152710&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2019.8849398

DO - 10.1109/ISIT.2019.8849398

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:85073152710

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 2304

EP - 2308

BT - 2019 IEEE International Symposium on Information Theory, ISIT 2019 - Proceedings

PB - Institute of Electrical and Electronics Engineers Inc.

Y2 - 7 July 2019 through 12 July 2019

ER -