Benign overfitting and adaptive nonparametric regression

Benign overfitting has attracted a great deal of attention in the recent years. It was initially motivated by the fact that deep neural networks have good predictive properties even when perfectly interpolating the training data [Belkin et al., 2019a], [Belkin et al., 2018b], [Zhang et al., 2021], [Belkin, 2021]. Such a behavior stands in strong contrast with the classical point of view that perfectly fitting the data points is not compatible with predicting well. With the aim of understanding this new phenomenon, a series of recent papers studied benign overfitting in linear regression setting, see [Bartlett et al., 2020], [Tsigler and Bartlett, 2020], [Chinot and Lerasle, 2020], [Muthukumar et al., 2020], [Bartlett and Long, 2021], [Lecué and Shang, 2022] and the references therein. The main conclusion for the linear model is that an unbalanced spectrum of the design matrix and over-parametrization, which in a sense approaches the model to non-parametric setting, are essential for benign overfitting to occur in linear regression. Extensions to kernel ridgeless regression were considered in [Liang and Rakhlin, 2020] when the sample size n and the dimension d were assumed to satisfy n null d, and in [Liang et al., 2020] for a more general case d null n