Data-sparse settings such as robotic manipulation, 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. Since traditional classification and regression error terms show up in the definitions of calibration error, it is natural to suspect that previous work can be used to help understand the relationship between equivariance and calibration error. In this work, we present a theory relating equivariance to uncertainty estimation. By proving lower and upper bounds on uncertainty calibration errors (ECE and ENCE) under various equivariance conditions, we elucidate the generalization limits of equivariant models and illustrate how symmetry mismatch can result in miscalibration in both classification and regression. We complement our theoretical framework with numerical experiments that clarify the relationship between equivariance and uncertainty using a variety of real and simulated datasets, and we comment on trends with symmetry mismatch, group size, and aleatoric and epistemic uncertainties.

翻译：数据稀疏场景（如机器人操控、分子物理学和星系形态分类）是深度学习最具挑战性的领域之一。针对这些问题，等变网络有助于改进对输入空间中欠采样区域的建模，而不确定性估计可防止模型过度自信。然而，迄今为止，等变性与模型置信度之间的关系，以及更广义的等变性与模型校准之间的关系尚未得到系统研究。由于传统分类和回归误差项出现在校准误差的定义中，自然可以推测先前的研究成果有助于理解等变性与校准误差之间的关联。本研究提出了一个将等变性与不确定性估计相联系的理论框架。通过证明在不同等变条件下不确定性校准误差（ECE和ENCE）的上下界，我们阐明了等变模型的泛化极限，并阐释了对称性失配如何导致分类和回归任务中的校准偏差。我们通过数值实验补充理论框架，利用真实与模拟数据集的多组实验阐明等变性与不确定性的关系，并对对称性失配、群组规模以及偶然性与认知性不确定性的影响趋势进行了分析。