We consider an underdetermined noisy linear regression model where the
minimum-norm interpolating predictor is known to be consistent, and ask: can
uniform convergence in a norm ball, or at least (following Nagarajan and
Kolter) the subset of a norm ball that the algorithm selects on a typical input
set, explain this success? We show that uniformly bounding the difference
between empirical and population errors cannot show any learning in the norm
ball, and cannot show consistency for any set, even one depending on the exact
algorithm and distribution. But we argue we can explain the consistency of the
minimal-norm interpolator with a slightly weaker, yet standard, notion: uniform
convergence of zero-error predictors in a norm ball. We use this to bound the
generalization error of low- (but not minimal-) norm interpolating predictors.

本研究在考虑一个欠定的，有噪音的线性回归模型，其中最小范数插值预测器因为一致性而被认为是有效的，研究者探究是否均匀收敛以及范数球时的学习机制能够解释其成功，发现虽然均匀收敛不能证明范数球中的学习，但可以用一种略弱但比较常见的方式解释最小范数插值器一致性的存在，同时该研究使用这种方式限定了低（但不是最小）范数插值器的泛化误差。