Implicit Regularization and Convergence for Weight Normalization

Normalization methods such as batch [Ioffe and Szegedy, 2015] , weight [Salimans and Kingma, 2016] , instance [Ulyanov et al., 2016] , and layer normalization [Ba et 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 2 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.