Network Degeneracy as an Indicator of Training Performance: Comparing Finite and Infinite Width Angle Predictions

Abstract

Neural networks are powerful functions with widespread use, but the theoretical behaviour of these functions is not fully understood. Creating deep neural networks by stacking many layers has achieved exceptional performance in many applications and contributed to the recent explosion of these methods. Previous works have shown that depth can exponentially increase the expressibility of the network. However, as networks get deeper and deeper, they are more susceptible to becoming degenerate. We observe this degeneracy in the sense that on initialization, inputs tend to become more and more correlated as they travel through the layers of the network. If a network has too many layers, it tends to approximate a (random) constant function, making it effectively incapable of distinguishing between inputs. This seems to affect the training of the network and cause it to perform poorly, as we empirically investigate in this paper. We use a simple algorithm that can accurately predict the level of degeneracy for any given fully connected ReLU network architecture, and demonstrate how the predicted degeneracy relates to training dynamics of the network. We also compare this prediction to predictions derived using infinite width networks.