Epistemic Uncertainty and Observation Noise with the Neural Tangent Kernel

TL;DR

This paper extends the neural tangent kernel framework to incorporate observation noise and provides a method to estimate epistemic uncertainty, demonstrated through synthetic regression experiments.

Contribution

It introduces a way to handle aleatoric noise and estimate epistemic uncertainty within the NTK framework, compatible with standard training procedures.

Findings

Effective handling of non-zero observation noise.

Provides an estimator for posterior covariance.

Validated through synthetic regression experiments.

Abstract

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.