The Loss Surface of Deep Linear Networks Viewed Through the Algebraic Geometry Lens

By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of deep linear neural network models. After providing clarification on the various definitions of “flat” minima, we show that the geometrically flat minima, which are merely artifacts of residual continuous symmetries of the deep linear networks, can be straightforwardly removed by a generalized
$L_2$
-regularization. Then, we establish upper bounds on the number of isolated stationary points of these networks with the help of algebraic geometry. Combining these upper bounds with a method in numerical algebraic geometry, we find
all
stationary points for modest depth and matrix size. We demonstrate that, in the presence of the non-zero regularization, deep linear networks can indeed possess local minima which are not global minima. Finally, we show that even though the number of stationary points increases as the number of neurons (regularization parameters) increases (decreases), higher index saddles are surprisingly rare.