Towards Quantifying the Hessian Structure of Neural Networks

Empirical studies reported that the Hessian matrix of neural networks (NNs) exhibits a near-block-diagonal structure, yet its theoretical foundation remains unclear. In this work, we reveal that the reported Hessian structure comes from a mixture of two forces: a ``static force'' rooted in the architecture design, and a ''dynamic force'' arisen from training. We then provide a rigorous theoretical analysis of ''static force'' at random initialization. We study linear models and 1-hidden-layer networks for classification tasks with $C$ classes. By leveraging random matrix theory, we compare the limit distributions of the diagonal and off-diagonal Hessian blocks and find that the block-diagonal structure arises as $C$ becomes large. Our findings reveal that $C$ is one primary driver of the near-block-diagonal structure. These results may shed new light on the Hessian structure of large language models (LLMs), which typically operate with a large $C$ exceeding $10^4$.