On Uncertainty Calibration for Invariant Functions

Data-sparse settings such as robotic manipulation, and molecular physics, and galaxy morphology classification are some of the hardest domains for deep learning. For these problems, equivariant networks can help improve modeling across undersampled parts of the input space, and uncertainty estimation can guard against overconfidence. However, until now, the relationships between equivariance and model confidence, and more generally equivariance and model calibration, has yet to be studied. In this work, we present the first theory relating invariance to uncertainty estimation. By proving lower and upper bounds on uncertainty calibration errors under various invariance conditions, we elucidate the generalization limits of invariant models.