Implicit Regularization and Convergence for Weight Normalization

TL;DR

This paper analyzes weight normalization and a variant called reparametrized projected gradient descent, showing they act as implicit regularizers that guide convergence towards minimum norm solutions in overparametrized least-squares regression, with less sensitivity to initialization.

Contribution

It demonstrates that weight normalization and rPGD implicitly regularize and converge near minimum norm solutions, differing from standard gradient descent behavior.

Findings

Weight normalization and rPGD regularize weights adaptively.

These methods converge close to minimum norm solutions.

They are less sensitive to initializations compared to gradient descent.

Abstract

Normalization methods such as batch [Ioffe and Szegedy, 2015], weight [Salimansand Kingma, 2016], instance [Ulyanov et al., 2016], and layer normalization [Baet al., 2016] have been widely used in modern machine learning. Here, we study the weight normalization (WN) method [Salimans and Kingma, 2016] and a variant called reparametrized projected gradient descent (rPGD) for overparametrized least-squares regression. WN and rPGD reparametrize the weights with a scale g and a unit vector w and thus the objective function becomes non-convex. We show that this non-convex formulation has beneficial regularization effects compared to gradient descent on the original objective. These methods adaptively regularize the weights and converge close to the minimum l2 norm solution, even for initializations far from zero. For certain stepsizes of g and w , we show that they can converge close to the minimum norm solution. This is different from the behavior of gradient descent, which converges to the minimum norm solution only when started at a point in the range space of the feature matrix, and is thus more sensitive to initialization.