Investigating the Overlooked Hessian Structure: From CNNs to LLMs

It is well-known that the Hessian of deep loss landscape matters to optimization and generalization of deep learning. Previous studies reported a rough Hessian structure in deep learning, which consists of two components, a small number of large eigenvalues and a large number of nearly-zero eigenvalues. To the best of our knowledge, we are the first to report that a simple but overlooked power-law Hessian structure exists in well-trained deep neural networks, including Convolutional Neural Networks (CNNs) and Large Language Models (LLMs). Moreover, we provide a maximum-entropy theoretical interpretation for the power-law Hessian structure and theoretically demonstrate the existence of a robust and low-dimensional subspace of deep neural networks. Our extensive experiments using the proposed power-law spectral method demonstrate that the power-law Hessian spectra critically relate to multiple important behaviors of deep learning, including optimization, generalization, and overparameterization. Notably, we discover that the power-law Hessian structure of a given LLM can often predict generalization during training in some occasions, while conventional sharpnessbased generalization measures which often work well on CNNs largely fail as an effective generalization predictor of LLMs. Dataset Model Training dks dc Power-Law MNIST LeNet Random 0.0796 0.0430 No MNIST LeNet SGD 0.00900 0.0430 Yes Fashion-MNIST LeNet Random 0.0971 0.0430 No Fashion-