Sharpness-Aware Minimization (SAM) has attracted significant attention for its effectiveness in improving generalization across various tasks. However, its underlying principles remain poorly understood. In this work, we analyze SAM's training dynamics using the maximum eigenvalue of the Hessian as a measure of sharpness, and propose a third-order stochastic differential equation (SDE), which reveals that the dynamics are driven by a complex mixture of second- and third-order terms. We show that alignment between the perturbation vector and the top eigenvector is crucial for SAM's effectiveness in regularizing sharpness, but find that this alignment is often inadequate in practice, limiting SAM's efficiency. Building on these insights, we introduce Eigen-SAM, an algorithm that explicitly aims to regularize the top Hessian eigenvalue by aligning the perturbation vector with the leading eigenvector. We validate the effectiveness of our theory and the practical advantages of our proposed approach through comprehensive experiments. Code is available at https://github.com/RitianLuo/EigenSAM.

Active learning, a widely adopted technique for enhancing machine learning models in text and image classification tasks with limited annotation resources, has received relatively little attention in the domain of Named Entity Recognition (NER). The challenge of data imbalance in NER has hindered the effectiveness of active learning, as sequence labellers lack sufficient learning signals. To address these challenges, this paper presents a novel reweighting-based active learning strategy that assigns dynamic smoothed weights to individual tokens. This adaptable strategy is compatible with various token-level acquisition functions and contributes to the development of robust active learners. Experimental results on multiple corpora demonstrate the substantial performance improvement achieved by incorporating our re-weighting strategy into existing acquisition functions, validating its practical efficacy.