  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.
