  For fixed training data and network parameters in the other layers the L1
loss of a ReLU neural network as a function of the first layer's parameters is
a piece-wise affine function. We use the Deep ReLU Simplex algorithm to
iteratively minimize the loss monotonically on adjacent vertices and analyze
the trajectory of these vertex positions. We empirically observe that in a
neighbourhood around a local minimum, the iterations behave differently such
that conclusions on loss level and proximity of the local minimum can be made
before it has been found: Firstly the loss seems to decay exponentially slow at
iterated adjacent vertices such that the loss level at the local minimum can be
estimated from the loss levels of subsequently iterated vertices, and secondly
we observe a strong increase of the vertex density around local minima. This
could have far-reaching consequences for the design of new gradient-descent
algorithms that might improve convergence rate by exploiting these facts.
