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 rate-optimal 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.

翻译：过参数化神经网络在训练误差趋近于零时的成功引发了人们对良性过拟合现象的极大兴趣，即尽管估计器插值了含噪训练数据，但统计上仍具有一致性。尽管固定维度下的良性过拟合已在某些学习方法中得到证实，但现有文献表明，对于典型核方法回归和宽神经网络，良性过拟合需要高维场景，即维度随样本量增长。本文表明，关键在于估计器的平滑性而非维度：良性过拟合当且仅当估计器的导数足够大时才可能实现。我们将现有不一致性结果推广至非插值模型及更多核方法，证明在固定维度下，中等导数的良性过拟合是不可能的。反之，我们证明对于具有大导数的尖峰-平滑核序列回归，可实现速率最优的良性过拟合。通过神经正切核，我们将结果推广至宽神经网络。我们证明，虽然使用ReLU激活的无限宽网络无法实现良性过拟合，但可通过在激活函数中添加微小的高频波动来修正。实验验证了此类神经网络在过拟合的同时，即使在低维数据集上也能良好泛化。