The success of over-parameterized neural networks trained to near-zero training error has caused great interest in the phenomenon of benign overfitting, where estimators are statistically consistent even though they interpolate noisy training data. While benign overfitting in fixed dimension has been established for some learning methods, current literature suggests that for regression with typical kernel methods and wide neural networks, benign overfitting requires a high-dimensional setting where the dimension grows with the sample size. In this paper, we show that the smoothness of the estimators, and not the dimension, is the key: benign overfitting is possible if and only if the estimator's derivatives are large enough. We generalize existing inconsistency results to non-interpolating models and more kernels to show that benign overfitting with moderate derivatives is impossible in fixed dimension. Conversely, we show that benign overfitting is possible for regression with a sequence of spiky-smooth kernels with large derivatives. Using neural tangent kernels, we translate our results to wide neural networks. We prove that while infinite-width networks do not overfit benignly with the ReLU activation, this can be fixed by adding small high-frequency fluctuations to the activation function. Our experiments verify that such neural networks, while overfitting, can indeed generalize well even on low-dimensional data sets.

本研究发现，神经网络的光滑度才是决定良性过拟合的关键，只有在评估器的导数充分大时才能实现良性过拟合。我们证明在固定维度中，光滑度适中的良性过拟合是不可能的，在回归模型中，采用一系列具有大导数的峰形平滑内核可以实现良性过拟合。通过添加小的高频波动到激活函数中，可以在无限宽的神经网络中实现良性过拟合，从而提高在低维数据集上的泛化性能。

固定维度下核函数和神经网络的良性过拟合