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The explicit incorporation of task-specific inductive biases through symmetryhas emerged as a general design precept in the development of high-performancemachine learning models. For example, group equivariant neural networks havedemonstrated impressive performance across various domains and applicationssuch as protein and drug design. A prevalent intuition about such models isthat 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 orbest-performing model is one where the model symmetry aligns with the datasymmetry. In this paper, we conduct a formal unified investigation of theseintuitions. To begin, we present general quantitative bounds that demonstratehow models capturing task-specific symmetries lead to improved generalization.In fact, our results do not require the transformations to be finite or evenform a group and can work with partial or approximate equivariance. Utilizingthis quantification, we examine the more general question of modelmis-specification i.e. when the model symmetries don't align with the datasymmetries. We establish, for a given symmetry group, a quantitative comparisonbetween the approximate/partial equivariance of the model and that of the datadistribution, precisely connecting model equivariance error and dataequivariance error. Our result delineates conditions under which the modelequivariance error is optimal, thereby yielding the best-performing model forthe given task and data.

Approximation-Generalization Trade-offs under (Approximate) Group Equivariance