Implicit Regularization of Gradient Flow on One-Layer Softmax Attention

We study gradient flow on the exponential loss for a classification problem with a one-layer softmax attention model, where the key and query weight matrices are trained separately. Under a separability assumption on the data, we show that when gradient flow achieves the minimal loss value, it further implicitly minimizes the nuclear norm of the product of the key and query weight matrices. Such implicit regularization can be described by a Support Vector Machine (SVM) problem with respect to the attention weights. This finding contrasts with prior results showing that the gradient descent induces an implicit regularization on the Frobenius norm on the product weight matrix when the key and query matrices are combined into a single weight matrix for training. For diagonal key and query matrices, our analysis builds upon the reparameterization technique and exploits approximate KKT conditions of the SVM associated with the classification data. Moreover, the results are extended to general weights configurations given proper alignment of the weight matrices' singular spaces with the data features at initialization.