Analyzing Finite Neural Networks: Can We Trust Neural Tangent Kernel Theory?
This work addresses a critical question for researchers and practitioners using NTK theory to understand or design finite-width neural networks, clarifying the conditions under which the theory holds and where it breaks down.
This paper empirically investigates the validity of Neural Tangent Kernel (NTK) theory for finite-width deep neural networks (DNNs), finding that its applicability depends on initialization hyperparameters and network depth. Specifically, NTK theory fails for deep networks with exploding gradients, where the kernel is random and changes significantly, while networks with vanishing gradients are in the NTK regime but become untrainable with depth.
Neural Tangent Kernel (NTK) theory is widely used to study the dynamics of infinitely-wide deep neural networks (DNNs) under gradient descent. But do the results for infinitely-wide networks give us hints about the behavior of real finite-width ones? In this paper, we study empirically when NTK theory is valid in practice for fully-connected ReLU and sigmoid DNNs. We find out that whether a network is in the NTK regime depends on the hyperparameters of random initialization and the network's depth. In particular, NTK theory does not explain the behavior of sufficiently deep networks initialized so that their gradients explode as they propagate through the network's layers: the kernel is random at initialization and changes significantly during training in this case, contrary to NTK theory. On the other hand, in the case of vanishing gradients, DNNs are in the the NTK regime but become untrainable rapidly with depth. We also describe a framework to study generalization properties of DNNs, in particular the variance of network's output function, by means of NTK theory and discuss its limits.