To address this gap, we study deterministic/stochastic versions of SAM with practical configurations (i.e., constant

As a significant direction in this line, hyperbolic representation learning has been shown to offer several advantages over conventional Euclidean geometry.

SAM aims to find a perturbed model within the vicinity of a current model that maximizes the loss over a training set.

Several techniques based on re-weighting and margin adjustment of loss are often used toenhance theperformance ofneural networks, particularly onminority classes.

Additionally, it is evident that the results in (ii) do not implytheconvergenceof f(xk) to0.

Graph Neural Networks (GNNs) have shown superior performance in node classification. However, GNNs perform poorly in the Few-Shot Node Classification (FSNC) task that requires robust generalization to make accurate predictions for unseen classes with limited labels. To tackle the challenge, we propose the integration of Sharpness-Aware Minimization (SAM)--a technique designed to enhance model generalization by finding a flat minimum of the loss landscape--into GNN training. The standard SAM approach, however, consists of two forward-backward steps in each training iteration, doubling the computational cost compared to the base optimizer (e.g., Adam). To mitigate this drawback, we introduce a novel algorithm, Fast Graph Sharpness-Aware Minimization (FGSAM), that integrates the rapid training of Multi-Layer Perceptrons (MLPs) with the superior performance of GNNs. Specifically, we utilize GNNs for parameter perturbation while employing MLPs to minimize the perturbed loss so that we can find a flat minimum with good generalization more efficiently.

Sharpness-Aware Minimization (SAM) is a recently proposed gradient-based optimizer (Foret et al., ICLR 2021) that greatly improves the prediction performance of deep neural networks. Consequently, there has been a surge of interest in explaining its empirical success.

Contemporary advances in the field of deep learning have embarked upon an exploration of the underlying geometric properties of data, thus encouraging the investigation of techniques that consider general manifolds, for example, hyperbolic or orthogonal neural networks. However, the optimization algorithms for training such geometric deep learning models still remain highly under-explored. In this paper, we introduce Riemannian SAM by generalizing conventional Euclidean SAM to Riemannian manifolds. We successfully formulate the sharpness-aware minimization on Riemannian manifolds, leading to one of a novel instantiation, Lorentz SAM. In addition, SAM variants proposed in previous studies such as Fisher SAM can be derived as special examples under our Riemannian SAM framework. We provide the convergence analysis of Riemannian SAM under a less aggressively decaying ascent learning rate than Euclidean SAM. Our analysis serves as a theoretically sound contribution encompassing a diverse range of manifolds, also providing the guarantees for SAM variants such as Fisher SAM, whose convergence analyses are absent. Lastly, we illustrate the superiority of Riemannian SAM in terms of generalization over previous Riemannian optimization algorithms through experiments on knowledge graph completion and machine translation tasks.