Convolutions encode equivariance symmetries into neural networks leading to better generalisation performance. However, symmetries provide fixed hard constraints on the functions a network can represent, need to be specified in advance, and can not be adapted. Our goal is to allow flexible symmetry constraints that can automatically be learned from data using gradients. Learning symmetry and associated weight connectivity structures from scratch is difficult for two reasons. First, it requires efficient and flexible parameterisations of layer-wise equivariances. Secondly, symmetries act as constraints and are therefore not encouraged by training losses measuring data fit. To overcome these challenges, we improve parameterisations of soft equivariance and learn the amount of equivariance in layers by optimising the marginal likelihood, estimated using differentiable Laplace approximations. The objective balances data fit and model complexity enabling layer-wise symmetry discovery in deep networks. We demonstrate the ability to automatically learn layer-wise equivariances on image classification tasks, achieving equivalent or improved performance over baselines with hard-coded symmetry.

翻译：卷积操作将等变对称性编码到神经网络中，从而提升泛化性能。然而，对称性对网络所能表示的函数施加了固定的硬约束，需要预先指定且无法自适应调整。本文旨在实现灵活的对称性约束，使其能够利用梯度从数据中自动学习。从头开始学习对称性及其关联的权重连接结构面临两大挑战：第一，需要高效且灵活的参数化方法来表征逐层等变性；第二，对称性作为约束条件，不会受到衡量数据拟合程度的训练损失的激励。为克服这些困难，我们改进了软等变性的参数化方法，并通过优化边际似然（利用可微拉普拉斯近似估计）来学习各层的等变性程度。该目标函数平衡了数据拟合与模型复杂度，使得深度网络能够逐层发现对称性。我们展示了在图像分类任务中自动学习逐层等变性的能力，其结果与采用硬编码对称性的基线方法相当或更优。