Title: Sharpness Can Be Manipulated and Misleading for Generalization
Primary area: optimization
Keywords: generalization, shaprness, hessian, activation sparsity, activation frequency, data augmentation

Abstract: Sharpness, commonly measured by the Hessian matrix, has long been hypothesized to correlate with generalization. However, this work presents several counterexamples where Hessian-based sharpness can be manipulated. We derive a formula for the Hessian trace, revealing its dependence on several key factors: the norm of network weights, activation frequency, and the entropy of the output distribution. By manipulating these factors, we construct scenarios where models reside in flat minima yet exhibit overfitting and poor test performance. Moreover, Gaussian noise injection  reduces Hessian trace within the first epoch and can even yield arbitrarily flat minima without corresponding improvement in generalization. This suggests that sharpness may be correlated with  generalization without being causally responsible for it.

TL;DR: This paper provides several counterexamples for the correlation between flat minima and generalization, proving that the sharpness metric can be manipulated.