Saddle-To-Saddle Dynamics in Deep ReLU Networks: Low-Rank Bias in the First Saddle Escape
Ioannis Bantzis, James B Simon, Arthur Jacot
Abstract
When a deep ReLU network is initialized with small weights, gradient descent (GD) is at first dominated by the saddle at the origin in parameter space. We study the so-called escape directions along which GD leaves the origin, which play a similar role as the eigenvectors of the Hessian for strict saddles. We show that the optimal escape direction features a \textit{low-rank bias} in its deeper layers: the first singular value of the $\ell$-th layer weight matrix is at least $\ell^{\frac{1}{4}}$ larger than any other singular value. We also prove a number of related results about these escape directions. We suggest that deep ReLU networks exhibit saddle-to-saddle dynamics, with GD visiting a sequence of saddles with increasing bottleneck rank.
BibTeX
@inproceedings{
bantzis2026saddletosaddle,
title={Saddle-To-Saddle Dynamics in Deep Re{LU} Networks: Low-Rank Bias in the First Saddle Escape},
author={Ioannis Bantzis and James B Simon and Arthur Jacot},
booktitle={The Fourteenth International Conference on Learning Representations},
year={2026},
url={https://openreview.net/forum?id=B4zcoLvjw0}
}