Kernel and Rich Regimes in Overparametrized Models

TL;DR

This paper investigates how the scale of initialization in overparametrized neural networks determines whether they operate in a kernel regime or a rich, active regime, affecting their generalization and implicit biases.

Contribution

It provides a detailed analysis of the transition between kernel and rich regimes controlled by initialization scale in simple and complex neural network models.

Findings

Transition between kernel and rich regimes depends on initialization scale.

Analysis of a simple two-layer model shows meaningful regime change.

Demonstrates the transition in matrix factorization and multilayer non-linear networks.

Abstract

A recent line of work studies overparametrized neural networks in the "kernel regime," i.e. when the network behaves during training as a kernelized linear predictor, and thus training with gradient descent has the effect of finding the minimum RKHS norm solution. This stands in contrast to other studies which demonstrate how gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. Building on an observation by Chizat and Bach, we show how the scale of the initialization controls the transition between the "kernel" (aka lazy) and "rich" (aka active) regimes and affects generalization properties in multilayer homogeneous models. We provide a complete and detailed analysis for a simple two-layer model that already exhibits an interesting and meaningful transition between the kernel and rich regimes, and we demonstrate the transition for more complex matrix factorization models and multilayer non-linear networks.