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 $y\in {\cal R}$ as a linear combination of the feature vector $x^T\theta$ where $x\in {\cal R}^M$, 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.

通过通用学习的角度重新审视线性回归问题，研究了标签y和特征向量x^T的线性组合表达式及其学习可行性，证明在训练数据的相关矩阵特征向量构成的子空间上，即使参数个数M多于样本个数N，线性回归也可以具有很好的推广性能。

一种新的旧问题视角：线性回归的通用学习方法