Learning (Approximately) Equivariant Networks via Constrained Optimization

Equivariant neural networks are designed to respect symmetries through their architecture, boosting generalization and sample efficiency when those symmetries are present in the data distribution. Real-world data, however, often departs from perfect symmetry because of noise, structural variation, measurement bias, or other symmetry-breaking effects. Strictly equivariant models may struggle to fit the data, while unconstrained models lack a principled way to leverage partial symmetries. Even when the data is fully symmetric, enforcing equivariance can hurt training by limiting the model to a restricted region of the parameter space. Guided by homotopy principles, where an optimization problem is solved by gradually transforming a simpler problem into a complex one, we introduce Adaptive Constrained Equivariance (ACE), a constrained optimization approach that starts with a flexible, non-equivariant model and gradually reduces its deviation from equivariance. This gradual tightening smooths training early on and settles the model at a data-driven equilibrium, balancing between equivariance and non-equivariance. Across multiple architectures and tasks, our method consistently improves performance metrics, sample efficiency, and robustness to input perturbations compared with strictly equivariant models and heuristic equivariance relaxations.