The practical success of overparameterized neural networks has motivated the
recent scientific study of interpolating methods, which perfectly fit their
training data. Certain interpolating methods, including neural networks, can
fit noisy training data without catastrophically bad test performance, in
defiance of standard intuitions from statistical learning theory. Aiming to
explain this, a body of recent work has studied benign overfitting, a
phenomenon where some interpolating methods approach Bayes optimality, even in
the presence of noise. In this work we argue that while benign overfitting has
been instructive and fruitful to study, many real interpolating methods like
neural networks do not fit benignly: modest noise in the training set causes
nonzero (but non-infinite) excess risk at test time, implying these models are
neither benign nor catastrophic but rather fall in an intermediate regime. We
call this intermediate regime tempered overfitting, and we initiate its
systematic study. We first explore this phenomenon in the context of kernel
(ridge) regression (KR) by obtaining conditions on the ridge parameter and
kernel eigenspectrum under which KR exhibits each of the three behaviors. We
find that kernels with powerlaw spectra, including Laplace kernels and ReLU
neural tangent kernels, exhibit tempered overfitting. We then empirically study
deep neural networks through the lens of our taxonomy, and find that those
trained to interpolation are tempered, while those stopped early are benign. We
hope our work leads to a more refined understanding of overfitting in modern
learning.

该论文研究了神经网络等插值方法是否能够在存在噪声的情况下，拟合训练数据而不会表现出灾难性的测试性能，尝试通过 “良性过拟合” 和 “温和过拟合” 两个现象进行解释，并首次系统研究了 “温和过拟合” 的性质及在核（岭）回归中的表现，以及在深度神经网络中的实验结果。