In this paper, we provide a theoretical study of noise geometry for minibatch stochastic gradient descent (SGD), a phenomenon where noise aligns favorably with the geometry of local landscape. We propose two metrics, derived from analyzing how noise influences the loss and subspace projection dynamics, to quantify the alignment strength. We show that for (over-parameterized) linear models and two-layer nonlinear networks, when measured by these metrics, the alignment can be provably guaranteed under conditions independent of the degree of over-parameterization. To showcase the utility of our noise geometry characterizations, we present a refined analysis of the mechanism by which SGD escapes from sharp minima. We reveal that unlike gradient descent (GD), which escapes along the sharpest directions, SGD tends to escape from flatter directions and cyclical learning rates can exploit this SGD characteristic to navigate more effectively towards flatter regions. Lastly, extensive experiments are provided to support our theoretical findings.

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