Bayesian Double Descent

Abstract

Double descent is a phenomenon of over-parameterized statistical models such as deep neural networks which have a re-descending property in their risk function. As the complexity of the model increases, risk exhibits a U-shaped region due to the traditional bias-variance trade-off, then as the number of parameters equals the number of observations and the model becomes one of interpolation where the risk can be unbounded and finally, in the over-parameterized region, it re-descends -- the double descent effect. Our goal is to show that this has a natural Bayesian interpretation. We also show that this is not in conflict with the traditional Occam's razor -- simpler models are preferred to complex ones, all else being equal. Our theoretical foundations use Bayesian model selection, the Dickey-Savage density ratio, and connect generalized ridge regression and global-local shrinkage methods with double descent. We illustrate our approach for high dimensional neural networks and provide detailed treatments of infinite Gaussian means models and non-parametric regression. Finally, we conclude with directions for future research.