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Analytic Insights into Structure and Rank of Neural Network Hessian Maps
Sidak Pal Singh · Gregor Bachmann · Thomas Hofmann

Thu Dec 09 08:30 AM -- 10:00 AM (PST) @ None #None

The Hessian of a neural network captures parameter interactions through second-order derivatives of the loss. It is a fundamental object of study, closely tied to various problems in deep learning, including model design, optimization, and generalization. Most prior work has been empirical, typically focusing on low-rank approximations and heuristics that are blind to the network structure. In contrast, we develop theoretical tools to analyze the range of the Hessian map, which provide us with a precise understanding of its rank deficiency and the structural reasons behind it. This yields exact formulas and tight upper bounds for the Hessian rank of deep linear networks --- allowing for an elegant interpretation in terms of rank deficiency. Moreover, we demonstrate that our bounds remain faithful as an estimate of the numerical Hessian rank, for a larger class of models such as rectified and hyperbolic tangent networks. Further, we also investigate the implications of model architecture (e.g.~width, depth, bias) on the rank deficiency. Overall, our work provides novel insights into the source and extent of redundancy in overparameterized neural networks.

Author Information

Sidak Pal Singh (ETH Zürich)
Gregor Bachmann (ETH Zürich)
Thomas Hofmann (ETH Zurich)

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