  The explicit incorporation of task-specific inductive biases through symmetry
has emerged as a general design precept in the development of high-performance
machine learning models. For example, group equivariant neural networks have
demonstrated impressive performance across various domains and applications
such as protein and drug design. A prevalent intuition about such models is
that the integration of relevant symmetry results in enhanced generalization.
Moreover, it is posited that when the data and/or the model may only exhibit
$\textit{approximate}$ or $\textit{partial}$ symmetry, the optimal or
best-performing model is one where the model symmetry aligns with the data
symmetry. In this paper, we conduct a formal unified investigation of these
intuitions. To begin, we present general quantitative bounds that demonstrate
how models capturing task-specific symmetries lead to improved generalization.
In fact, our results do not require the transformations to be finite or even
form a group and can work with partial or approximate equivariance. Utilizing
this quantification, we examine the more general question of model
mis-specification i.e. when the model symmetries don't align with the data
symmetries. We establish, for a given symmetry group, a quantitative comparison
between the approximate/partial equivariance of the model and that of the data
distribution, precisely connecting model equivariance error and data
equivariance error. Our result delineates conditions under which the model
equivariance error is optimal, thereby yielding the best-performing model for
the given task and data.
