Explaining f-Divergence-Based Regularization via Local Curvature and Sharpness-Aware Minimization

Sharpness-Aware Minimization (SAM) improves generalization by minimizing the worst-case loss within a fixed parameter-space radius neighborhood. SAM and its variants mainly rely on a first-order linearized surrogate, while flat minima are inherently a second-order (curvature) notion.We revisit this mismatch and propose Loss-Equated SAM (LE-SAM), which inverts the traditional SAM mechanism that fixed perturbation radius with a fixed loss-space budget,effectively removing gradient-norm-dominated learning signals and shifting optimization toward curvature-dominated terms. Extensive experiments across diverse benchmarks and tasks demonstrate the strong generalization ability of LESAM that consistently outperforms SAM and even its variants, achieving the state-of-the-art performance.

A widely held intuition in deep learning is that stochastic gradient descent (SGD) implicitly favors flat minima and that flat minima generalize better, but standard Euclidean measures of flatness such as the trace or maximum eigenvalue of the loss Hessian are not invariant under reparametrizations that preserve the network function, which undermines the theoretical foundations of this narrative. In this study we resolve this issue by grounding flatness in the Riemannian geometry of the statistical manifold induced by the Fisher Information Matrix (FIM). We define Riemannian sharpness mathematically and prove that it is invariant under smooth, function-preserving reparametrizations, which directly addresses the critique of Dinh et al. in the paper ``Sharp minima can generalize for deep nets''.We note that this invariance is a property of the true FIM; the diagonal empirical estimator used in practice (and in all experiments below) inherits invariance only approximately, and exact invariance under arbitrary reparametrizations would require structured estimators such as K-FAC. We formalize the gradient noise of mini-batch SGD as having a covariance structure proportional to the FIM, derive the stationary distribution of the resulting stochastic differential equation, and then show that the probability mass is exponentially concentrated at Riemannian-flat minima. A PAC-Bayes generalization bound controlled explicitly by SR formally links this geometric bias to test performance. Our experiments on MNIST and CIFAR-10 confirm that SR reliably tracks generalization in ways that Euclidean sharpness does not, and that its scaling with $η/B$ matches the theoretical predictions. Together these results provide a rigorous, reparametrization-invariant account of why flat minima generalize.

Sharpness-Aware Minimization (SAM) improves generalization by seeking parameters whose loss is robust to local adversarial perturbations, but the quantitative mechanism underlying its implicit bias toward flat minima remains unclear. In particular, the perturbation radius $ρ$ is typically treated as an isolated tuning parameter, despite defining the neighborhood in which SAM measures sharpness. We analyze mini-batch SAM near an interpolating minimum through linear stability. Under local linearization and gradient-noise alignment assumptions, we prove that every linearly stable minimum satisfies $λ_{\max}\leq\sqrt[3]{bΓ/(2ρη^2)}$, where $λ_{\max}$ is the largest Hessian eigenvalue, $b$ is the batch size, $η$ is the learning rate, and $Γ$ bounds the gradient norm. The bound quantitatively characterizes SAM's implicit flatness bias: holding the other quantities fixed, a smaller batch size, a larger learning rate, or a larger radius restricts linearly stable SAM to flatter minima. It also exposes a necessary trade-off: $ρ$ should be large enough to promote flatness, yet remain local enough to preserve the approximation and stable training. We validate this prediction in a controlled study of 900 models on CIFAR-100 with ResNet-18 and VGG-19, where increasing $ρ$ is consistently associated with a smaller largest Hessian eigenvalue across batch-size and learning-rate settings. Finally, we instantiate the analysis in Taylor-Locality Controlled SAM (TLC-SAM), which adjusts $ρ$ using the observed Taylor-approximation error and further reduces the top Hessian eigenvalue relative to fixed-radius SAM. Our results provide quantitative hyperparameter bounds and a stability--locality perspective for analyzing and designing SAM variants.