Computer Science > Machine Learning
[Submitted on 8 Oct 2020 (v1), last revised 21 Oct 2022 (this version, v6)]
Title:Dissecting Hessian: Understanding Common Structure of Hessian in Neural Networks
View PDFAbstract:Hessian captures important properties of the deep neural network loss landscape. Previous works have observed low rank structure in the Hessians of neural networks. In this paper, we propose a decoupling conjecture that decomposes the layer-wise Hessians of a network as the Kronecker product of two smaller matrices. We can analyze the properties of these smaller matrices and prove the structure of top eigenspace random 2-layer networks. The decoupling conjecture has several other interesting implications - top eigenspaces for different models have surprisingly high overlap, and top eigenvectors form low rank matrices when they are reshaped into the same shape as the corresponding weight matrix. All of these can be verified empirically for deeper networks. Finally, we use the structure of layer-wise Hessian to get better explicit generalization bounds for neural networks.
Submission history
From: Xingyu Zhu [view email][v1] Thu, 8 Oct 2020 21:18:11 UTC (11,521 KB)
[v2] Fri, 30 Oct 2020 18:02:21 UTC (11,521 KB)
[v3] Mon, 30 Nov 2020 08:44:47 UTC (11,519 KB)
[v4] Fri, 19 Feb 2021 15:39:47 UTC (13,651 KB)
[v5] Wed, 16 Jun 2021 15:27:49 UTC (20,218 KB)
[v6] Fri, 21 Oct 2022 17:23:00 UTC (20,298 KB)
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