  In this paper, we study the implicit regularization of stochastic gradient
descent (SGD) through the lens of {\em dynamical stability} (Wu et al., 2018).
We start by revising existing stability analyses of SGD, showing how the
Frobenius norm and trace of Hessian relate to different notions of stability.
Notably, if a global minimum is linearly stable for SGD, then the trace of
Hessian must be less than or equal to $2/\eta$, where $\eta$ denotes the
learning rate. By contrast, for gradient descent (GD), the stability imposes a
similar constraint but only on the largest eigenvalue of Hessian. We then turn
to analyze the generalization properties of these stable minima, focusing
specifically on two-layer ReLU networks and diagonal linear networks. Notably,
we establish the {\em equivalence} between these metrics of sharpness and
certain parameter norms for the two models, which allows us to show that the
stable minima of SGD provably generalize well. By contrast, the
stability-induced regularization of GD is provably too weak to ensure
satisfactory generalization. This discrepancy provides an explanation of why
SGD often generalizes better than GD. Note that the learning rate (LR) plays a
pivotal role in the strength of stability-induced regularization. As the LR
increases, the regularization effect becomes more pronounced, elucidating why
SGD with a larger LR consistently demonstrates superior generalization
capabilities. Additionally, numerical experiments are provided to support our
theoretical findings.
