On the Epistemic Uncertainty of Overparametrized Neural Networks

Epistemic uncertainty is often viewed as a reducible uncertainty that vanishes with increasing data. This perspective implicitly assumes parameter identifiability and equates epistemic uncertainty with predictive variability. In overparametrized neural networks, however, model parameters are typically non-identifiable due to symmetries and redundant representations. As a consequence, substantial parameter uncertainty can persist even when the underlying function is fully identified. In this work, we analyze epistemic uncertainty through the lens of non-identifiability and characterize both discrete and continuous sources of residual uncertainty. Focusing on one-hidden-layer ReLU networks, we thoroughly analyze the resulting posterior structure and validate our theoretical insights through empirical studies.

Many machine learning systems not only make predictions, but also try to estimate how uncertain they are. This uncertainty is often assumed to disappear as more data become available. However, modern neural networks can have many different internal parameter settings that represent exactly the same prediction function. This means that the network may become very certain about its predictions while still remaining uncertain about its internal parameters. This paper studies that phenomenon for overparametrized neural networks, focusing on simple ReLU networks where the mathematical structure can be analyzed precisely. It shows that some uncertainty comes from familiar symmetries, such as permuting hidden neurons, while additional continuous uncertainty arises when redundant neurons can split the role of a true neuron.