The Implicit Bias of Gradient Noise: A Symmetry Perspective

We characterize the learning dynamics of stochastic gradient descent (SGD) when continuous symmetry exists in the loss function, where the divergence between SGD and gradient descent is dramatic. We show that depending on how the symmetry affects the learning dynamics, we can divide a family of symmetry into two classes. For one class of symmetry, SGD naturally converges to solutions that have a balanced and aligned gradient noise. For the other class of symmetry, SGD will almost always diverge. Then, we show that our result remains applicable and can help us understand the training dynamics even when the symmetry is not present in the loss function. Our main result is universal in the sense that it only depends on the existence of the symmetry and is independent of the details of the loss function. We demonstrate that the proposed theory offers an explanation of progressive sharpening and flattening and can be applied to common practical problems such as representation normalization, matrix factorization, and the use of warmup.