Asymptotics of Ridge Regression in Convolutional Models

TL;DR

This paper analyzes the asymptotic behavior of ridge regression estimators in convolutional linear models, revealing the double descent phenomenon and providing exact error formulas in high-dimensional settings.

Contribution

It provides the first detailed asymptotic analysis of ridge estimators for convolutional models, including exact error formulas and empirical validation of double descent.

Findings

Double descent phenomenon observed in convolutional models

Exact asymptotic formulas for ridge estimation error derived

Theoretical results match experimental observations

Abstract

Understanding generalization and estimation error of estimators for simple models such as linear and generalized linear models has attracted a lot of attention recently. This is in part due to an interesting observation made in machine learning community that highly over-parameterized neural networks achieve zero training error, and yet they are able to generalize well over the test samples. This phenomenon is captured by the so called double descent curve, where the generalization error starts decreasing again after the interpolation threshold. A series of recent works tried to explain such phenomenon for simple models. In this work, we analyze the asymptotics of estimation error in ridge estimators for convolutional linear models. These convolutional inverse problems, also known as deconvolution, naturally arise in different fields such as seismology, imaging, and acoustics among others. Our results hold for a large class of input distributions that include i.i.d. features as a special case. We derive exact formulae for estimation error of ridge estimators that hold in a certain high-dimensional regime. We show the double descent phenomenon in our experiments for convolutional models and show that our theoretical results match the experiments.