Why Transformers Need Adam: A Hessian Perspective

SGD performs worse than Adam by a significant margin on Transformers, but the reason remains unclear. In this work, we provide an explanation of SGD's bad performance on Transformers through the lens of Hessian: (i) Transformers are "heterogeneous": the Hessian spectrum across parameter blocks vary dramatically, a phenomenon we call "block heterogeneity"; (ii) Heterogeneity hampers SGD: SGD performs badly on problems with block heterogeneity. To validate that heterogeneity hampers SGD, we check various Transformers, CNNs, MLPs, and quadratic problems, and find that SGD works well on problems without block heterogeneity but performs badly when the heterogeneity exists. Our initial theoretical analysis indicates that SGD performs poorly because it applies one single learning rate to all blocks, which cannot handle the heterogeneity among blocks. This limitation could be ameliorated if we use coordinate-wise learning rates, as designed in Adam.