Neural Tangent Kernel Beyond the Infinite-Width Limit: Effects of Depth and Initialization

Neural Tangent Kernel (NTK) is widely used to analyze overparametrized neural networks due to the famous result by (Jacot et al., 2018): in the infinite-width limit, the NTK is deterministic and constant during training. However, this result cannot explain the behavior of deep networks, since it generally does not hold if depth and width tend to infinity simultaneously. In this paper, we study the NTK of fully-connected ReLU networks with depth comparable to width. We prove that the NTK properties depend significantly on the depth-to-width ratio and the distribution of parameters at initialization. In fact, our results indicate the importance of the three phases in the hyperparameter space identified in (Poole et al., 2016): ordered, chaotic and the edge of chaos (EOC). We derive exact expressions for the NTK dispersion in the infinite-depth-and-width limit in all three phases and conclude that the NTK variability grows exponentially with depth at the EOC and in the chaotic phase but not in the ordered phase. We also show that the NTK of deep networks may stay constant during training only in the ordered phase and discuss how the structure of the NTK matrix changes during training.