  Recent works show that including group equivariance as an inductive bias
improves neural network performance for both classification and generation
tasks. Designing group-equivariant neural networks is, however, challenging
when the group of interest is large and is unknown. Moreover, inducing
equivariance can significantly reduce the number of independent parameters in a
network with fixed feature size, affecting its overall performance. We address
these problems by proving a new group-theoretic result in the context of
equivariant neural networks that shows that a network is equivariant to a large
group if and only if it is equivariant to smaller groups from which it is
constructed. We also design an algorithm to construct equivariant networks that
significantly improves computational complexity. Further, leveraging our
theoretical result, we use deep Q-learning to search for group equivariant
networks that maximize performance, in a significantly reduced search space
than naive approaches, yielding what we call autoequivariant networks (AENs).
To evaluate AENs, we construct and release new benchmark datasets, G-MNIST and
G-Fashion-MNIST, obtained via group transformations on MNIST and Fashion-MNIST
respectively. We show that AENs find the right balance between group
equivariance and number of parameters, thereby consistently having good task
performance.
