  We characterize regions of a loss surface as corridors when the continuous
curves of steepest descent -- the solutions of the gradient flow -- become
straight lines. We show that corridors provide insights into gradient-based
optimization, since corridors are exactly the regions where gradient descent
and the gradient flow follow the same trajectory, while the loss decreases
linearly. As a result, inside corridors there are no implicit regularization
effects or training instabilities that have been shown to occur due to the
drift between gradient descent and the gradient flow. Using the loss linear
decrease on corridors, we devise a learning rate adaptation scheme for gradient
descent; we call this scheme Corridor Learning Rate (CLR). The CLR formulation
coincides with a special case of Polyak step-size, discovered in the context of
convex optimization. The Polyak step-size has been shown recently to have also
good convergence properties for neural networks; we further confirm this here
with results on CIFAR-10 and ImageNet.
