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.

翻译：等变神经网络旨在通过其架构设计来保持对称性，当数据分布中存在这些对称性时，能够提升泛化能力和样本效率。然而，由于噪声、结构变异、测量偏差或其他对称性破缺效应，现实世界的数据往往偏离完美对称性。严格等变模型可能难以拟合数据，而无约束模型则缺乏利用部分对称性的原则性方法。即使数据完全对称，强制等变性也可能通过将模型限制在参数空间的受限区域而损害训练效果。受同伦原理（即通过逐步将简单问题转化为复杂问题来求解优化问题）的启发，我们引入了自适应约束等变性（ACE），这是一种约束优化方法，从灵活的非等变模型出发，逐步减小其与等变性的偏离。这种逐步收紧的策略在训练早期平滑了优化过程，并使模型收敛于数据驱动的平衡点，在等变性与非等变性之间取得权衡。在多种架构和任务中，与严格等变模型及启发式等变性松弛方法相比，我们的方法在性能指标、样本效率和输入扰动鲁棒性方面均表现出持续改进。