Symmetry Leads to Structured Constraint of Learning

Due to common architecture designs, symmetries exist extensively in contemporary neural networks. In this work, we unveil the importance of the loss function symmetries in affecting, if not deciding, the learning behavior of machine learning models. We prove that every mirror symmetry of the loss function leads to a structured constraint, which becomes a favored solution when either the weight decay or gradient noise is large. As direct corollaries, we show that rescaling symmetry leads to sparsity, rotation symmetry leads to low rankness, and permutation symmetry leads to homogeneous ensembling. Then, we show that the theoretical framework can explain the loss of plasticity and various collapse phenomena in neural networks and suggest how symmetries can be used to design algorithms to enforce hard constraints in a differentiable way.