The loss surface of deep linear networks viewed through the algebraic geometry lens
This work provides incremental insights into the optimization landscapes of deep linear networks, which is a foundational problem in machine learning for researchers studying neural network training dynamics.
The paper tackled the problem of understanding the loss surface of deep linear neural networks by applying algebraic geometry to characterize stationary points and flat minima, showing that with non-zero regularization, these networks can have local minima that are not global minima.
By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of the deep linear neural network models. After clarifying 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. Using these upper bounds and utilizing a numerical algebraic geometry method, we find all stationary points of modest depth and matrix size. We show that in the presence of the non-zero regularization, deep linear networks indeed possess local minima which are not the global minima. Our computational results clarify certain aspects of the loss surfaces of deep linear networks and provide novel insights.