Epistemic Uncertainty and Observation Noise with the Neural Tangent Kernel

Recent work has shown that training wide neural networks with gradient descent is formally equivalent to computing the mean of the posterior distribution in a Gaussian Process (GP) with the Neural Tangent Kernel (NTK) as the prior covariance and zero aleatoric noise \parencite{jacot2018neural}. In this paper, we extend this framework in two ways. First, we show how to deal with non-zero aleatoric noise. Second, we derive an estimator for the posterior covariance, giving us a handle on epistemic uncertainty. Our proposed approach integrates seamlessly with standard training pipelines, as it involves training a small number of additional predictors using gradient descent on a mean squared error loss. We demonstrate the proof-of-concept of our method through empirical evaluation on synthetic regression.