Noise Stability Optimization for Flat Minima with Optimal Convergence Rates

We consider finding flat, local minimizers by adding average weight perturbations. Given a nonconvex function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ and a $d$-dimensional distribution $\mathcal{P}$ which is symmetric at zero, we perturb the weight of $f$ and define $F(W) = \mathbb{E}[f({W + U})]$, where $U$ is a random sample from $\mathcal{P}$. This injection induces regularization through the Hessian trace of $f$ for small, isotropic Gaussian perturbations. Thus, the weight-perturbed function biases to minimizers with low Hessian trace. Several prior works have studied settings related to this weight-perturbed function by designing algorithms to improve generalization. Still, convergence rates are not known for finding minima under the average perturbations of the function $F$. This paper considers an SGD-like algorithm that injects random noise before computing gradients while leveraging the symmetry of $\mathcal{P}$ to reduce variance. We then provide a rigorous analysis, showing matching upper and lower bounds of our algorithm for finding an approximate first-order stationary point of $F$ when the gradient of $f$ is Lipschitz-continuous. We empirically validate our algorithm for several image classification tasks with various architectures. Compared to sharpness-aware minimization, we note a 12.6% and 7.8% drop in the Hessian trace and top eigenvalue of the found minima, respectively, averaged over eight datasets. Ablation studies validate the benefit of the design of our algorithm.