  Datasets often have their intrinsic symmetries, and particular deep-learning
models called equivariant or invariant models have been developed to exploit
these symmetries. However, if some or all of these symmetries are only
approximate, which frequently happens in practice, these models may be
suboptimal due to the architectural restrictions imposed on them. We tackle
this issue of approximate symmetries in a setup where symmetries are mixed,
i.e., they are symmetries of not single but multiple different types and the
degree of approximation varies across these types. Instead of proposing a new
architectural restriction as in most of the previous approaches, we present a
regularizer-based method for building a model for a dataset with mixed
approximate symmetries. The key component of our method is what we call
equivariance regularizer for a given type of symmetries, which measures how
much a model is equivariant with respect to the symmetries of the type. Our
method is trained with these regularizers, one per each symmetry type, and the
strength of the regularizers is automatically tuned during training, leading to
the discovery of the approximation levels of some candidate symmetry types
without explicit supervision. Using synthetic function approximation and motion
forecasting tasks, we demonstrate that our method achieves better accuracy than
prior approaches while discovering the approximate symmetry levels correctly.
