Issues with Neural Tangent Kernel Approach to Neural Networks

Neural tangent kernels (NTKs) have been proposed to study the behavior of trained neural networks from the perspective of Gaussian processes. An important result in this body of work is the theorem of equivalence between a trained neural network and kernel regression with the corresponding NTK. This theorem allows for an interpretation of neural networks as special cases of kernel regression. However, does this theorem of equivalence hold in practice? In this paper, we revisit the derivation of the NTK rigorously and conduct numerical experiments to evaluate this equivalence theorem. We observe that adding a layer to a neural network and the corresponding updated NTK do not yield matching changes in the predictor error. Furthermore, we observe that kernel regression with a Gaussian process kernel in the literature that does not account for neural network training produces prediction errors very close to that of kernel regression with NTKs. These observations suggest the equivalence theorem does not hold well in practice and puts into question whether neural tangent kernels adequately address the training process of neural networks.

Guiding Neural Network Initialization via Marginal Likelihood Maximization

We propose a simple, data-driven approach to help guide hyperparameter selection for neural network initialization. We leverage the relationship between neural network and Gaussian process models having corresponding activation and covariance functions to infer the hyperparameter values desirable for model initialization. Our experiment shows that marginal likelihood maximization provides recommendations that yield near-optimal prediction performance on MNIST classification task under experiment constraints. Furthermore, our empirical results indicate consistency in the proposed technique, suggesting that computation cost for the procedure could be significantly reduced with smaller training sets.