  Linear interpolation between initial neural network parameters and converged
parameters after training with stochastic gradient descent (SGD) typically
leads to a monotonic decrease in the training objective. This Monotonic Linear
Interpolation (MLI) property, first observed by Goodfellow et al. (2014)
persists in spite of the non-convex objectives and highly non-linear training
dynamics of neural networks. Extending this work, we evaluate several
hypotheses for this property that, to our knowledge, have not yet been
explored. Using tools from differential geometry, we draw connections between
the interpolated paths in function space and the monotonicity of the network -
providing sufficient conditions for the MLI property under mean squared error.
While the MLI property holds under various settings (e.g. network architectures
and learning problems), we show in practice that networks violating the MLI
property can be produced systematically, by encouraging the weights to move far
from initialization. The MLI property raises important questions about the loss
landscape geometry of neural networks and highlights the need to further study
their global properties.
