When a deep ReLU network is initialized with small weights, GD is at first dominated by the saddle at the origin in parameter space. We study the so-called escape directions, which play a similar role as the eigenvectors of the Hessian for strict saddles. We show that the optimal escape direction features a 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 argue that this result is a first step in proving Saddle-to-Saddle dynamics in deep ReLU networks, where GD visits a sequence of saddles with increasing bottleneck rank.

Empirical studies have demonstrated that the noise in stochastic gradient descent (SGD) aligns favorably with the local geometry of loss landscape. However, theoretical and quantitative explanations for this phenomenon remain sparse. In this paper, we offer a comprehensive theoretical investigation into the aforementioned {\em noise geometry} for over-parameterized linear (OLMs) models and two-layer neural networks. We scrutinize both average and directional alignments, paying special attention to how factors like sample size and input data degeneracy affect the alignment strength. As a specific application, we leverage our noise geometry characterizations to study how SGD escapes from sharp minima, revealing that the escape direction has significant components along flat directions. This is in stark contrast to GD, which escapes only along the sharpest directions. To substantiate our theoretical findings, both synthetic and real-world experiments are provided.