We analyze the landscape and training dynamics of diagonal linear networks in a linear regression task, with the network parameters being perturbed by isotropic normal noise during training. The addition of such noise may be interpreted as a stochastic form of sharpness-aware minimization (SAM) and we prove several results that relate its action on the underlying landscape and training dynamics to the sharpness of the loss. In particular, the noise induces a weighted mixture of fractional norm penalties on the network parameters, which forces the individual layers to balance at a fast rate and changes the underlying landscape to favor solutions that result from a shrinkage-thresholding operator applied to the true parameter. We show that balancing the layers equates to minimizing the average sharpness, as well as the trace of the Hessian matrix, among all possible factorizations of the same linear predictor. Further, we characterize how the noise level of the normal perturbations acts as a regularization parameter, with exact descriptions of its effect on shrinkage, thresholding, and balancing speed.

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