Pretraining accounts for a large fraction of the total computational cost in LLM training. However, noise-dominant gradients and the highly ill-conditioned loss landscape bring severe challenges. Although modern adaptive optimizers such as AdamW and Muon have achieved great success in large-scale pretraining, their reliance on gradient normalization offers limited mitigation of the ill-conditioned curvature. The progress along flat directions (eigen-directions of small eigenvalues), which dominates the final loss reduction, remains relatively slow. To enhance training dynamics along flat directions, we propose a curvature-conditioned multiscale momentum method with sphere constraints, delivering steady acceleration in LLM pretraining. This multiscale momentum, applied only along flat directions, pairs a slow-decay component for noise reduction with a fast-decay component for rapid curvature adaptation, harnessing their complementary strengths. Crucially, we employ a sphere constraint technique to prevent parameter inflation and excessively rapid effective learning rate decay that would otherwise arise from a naive combination. Extensive experiments show that the proposed method significantly accelerates Muon across diverse architectures (dense, MoE) and model sizes (0.12B--2.3B parameters). Theoretically, we verify the acceleration effect and provide insight into the design principles underlying the flat-direction multiscale momentum.
Muon's polar update is exact for an unweighted spectral geometry. We introduce GO-MUON, which uses a matched data-dependent geometry and reuses it across several optimization steps. Conditioned on any positive-definite left and right maps, its raw update exactly solves the corresponding weighted spectral oracle; this statement is independent of how the maps are estimated or how recently they were refreshed. For softmax cross-entropy, we quantify when the observed-label backward factor approaches the model Fisher and generalized Gauss--Newton factor. We also show that four-step refresh nearly preserves the tracking delay of slowly changing geometry while increasing stationary factor noise, making lazy geometry a compute--statistics tradeoff rather than a denoising mechanism.
Networks provide a fundamental representation of relationships among entities. However, real-world networks are often corrupted by noise caused by measurement errors and inherent stochasticity, hindering the discovery of meaningful structure. Most denoising methods rely on similarity-driven diffusion and ignore the non-Euclidean geometry of graphs, where local variations induce heterogeneous information transport. This motivates a geometric revisit of network denoising. In this work, we propose Ricci-Diffusion, a curvature-guided graph diffusion method inspired by Ricci flow. Specifically, Ricci-Diffusion exhibits a Ricci-flow-like evolution, in which relative edge-level curvature modulates local transport in the diffusion kernel and guides edge-weight updates toward a more regular graph geometry. We further provide a theoretical analysis showing that curvature can distinguish graph structures that common similarity-driven diffusion kernels fail to separate, and that curvature induces first-order corrections in one-step diffusion updates. The resulting diffusion process explicitly characterizes transport heterogeneity across local geometries and admits theoretical convergence to a stable denoised network. Results on real-world and synthetic graphs show that curvature-guided updates and curvature homogenization improve structure recovery and downstream performance.
Curvature notions on graphs, particularly Ollivier-Ricci and Forman, have emerged as powerful tools for addressing fundamental issues in Graph Neural Networks (GNNs) such as oversmoothing and oversquashing, but rely almost exclusively on local edge-level comparisons and therefore fail to certify how information actually propagates over long distances. We introduce Entropic Curvature, a global, transport-based curvature obtained by extending the Lott-Sturm-Villani framework to graphs through the displacement convexity of entropy along Wasserstein geodesics. We define a tractable Weak Entropic Curvature proxy that lower-bounds the global entropic curvature, and from it derive (i) a Poincare-type inequality controlling oversmoothing, (ii) a transport-entropy generalization bound, and (iii) an expansion paradox proving that sparsity, strong spectral expansion, and positive entropic curvature cannot coexist in large graphs, unifying oversmoothing and oversquashing as opposite ends of a single curvature spectrum. We translate the theory into three practical mechanisms, the E-Gate aggregator, the ENT structural encoding, and Midpoint-Completion Rewiring (MCR), and benchmark them against SDRF, FoSR, BORF, LCP, and Graph Ricci Flow on six node-classification benchmarks, and graph-classification.
Detecting communities in heterophilic graphs -- where connected nodes often belong to different classes -- is hard for unsupervised methods: classical modularity and spectral methods are feature agnostic, while deep graph-clustering methods rely on contrastive or generative machinery that is opaque. We propose Curvature-Guided Sheaf Diffusion (CGSD), a fully unsupervised community-detection algorithm that uses the discrete Forman--Ricci curvature of each edge as its single topological signal, propagated through every stage of an end-to-end pipeline. CGSD makes three concrete contributions: (i)~a curvature-gated sheaf-diffusion encoder that gates edge messages by $σ(κ_e)$ and is trained from three label-free structural losses (modularity, anti-collapse, curvature-weighted reconstruction); (ii)~a curvature-aware spectral clusterer (CSpec) that re-weights the $k$-NN affinity of the embedding by $σ(ακ_{e^*})$ before Ng--Jordan--Weiss; and (iii)~a unified label-free evaluation against nine truly-unsupervised baselines. On five heterophilic benchmarks (Cora, Cornell, Texas, Wisconsin, Chameleon), CGSD wins outright on Wisconsin and Chameleon and is competitive on the remaining three against nine unsupervised baselines. The gain over the strongest baseline is driven by the clusterer, not the encoder: on the same embedding, CSpec improves mean NMI from $0.091$ with $K$-Means to $0.107$ ($+15\%$, paired $t$-test $p=0.008$). The mechanism is interpretable: intra-community and inter-community curvature distributions are visibly separated. Code is open-sourced at https://github.com/woodywff/cgsd.
Current evaluation practices in relational learning rely heavily on flat leaderboards that average performance across heterogeneous datasets, implicitly assuming a uniform underlying structure. We show that this assumption introduces systematic bias: it obscures geometry-dependent performance variations and can lead to misleading conclusions about model generalization. In this work, we identify intrinsic geometry as a key latent factor governing model effectiveness. We demonstrate that conventional aggregated metrics mask critical performance trade-offs that only become visible when datasets are stratified by their geometric properties. To address this issue, we introduce a curvature-stratified evaluation framework that partitions datasets into positive, negative, and near-zero curvature regimes. Our benchmark evaluates 18 representative models including Graph Convolutional Networks (GCNs), Graph Foundation Models (GFMs), and tabular learning methods across 14 datasets. We find that model rankings are highly stable within each curvature regime but shift significantly across regimes, indicating that performance is fundamentally geometry-dependent rather than universally transferable. Notably, we identify regimes where GFMs offer diminishing returns compared to geometry-aligned GNNs. Based on these findings, we propose a geometry-aware evaluation protocol that yields more reliable and interpretable comparisons than standard aggregated benchmarks. We release all code, curvature-stratified dataset splits, and evaluation tools to support reproducible and rigorous assessment of future relational learning methods. Code and datasets are provided in our project homepage: https://sirbabbage.github.io/CurvBench_HOME/.
Shuche Wang, Fengzhuo Zhang, Jiaxiang Li +2cs.LG cs.AI
Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this curvature penalty into the squared update norm and Normalized Directional Sharpness (NDS). We find that Muon and Adam have comparable update norms, so Muon's smaller curvature penalty is driven by lower NDS, not update scale. Third, we study how training data and model structure shape Muon's NDS advantage. Using Zipf-Probabilistic Context-Free Grammar (PCFG) data with controlled imbalance, we show that data imbalance amplifies Muon's NDS advantage over Adam. A within-/cross-layer decomposition further shows that, in the middle and late stages of training, Muon's lower NDS is mainly sustained by smaller within-layer curvature. Beyond empirical evidence, we analyze stylized quadratic problems with heterogeneous curvature and gradient alignment toward high-curvature modes. We prove that Muon attains a smaller average NDS than GD by balancing update energy across curvature groups; when curvature heterogeneity is sufficiently strong, this also yields lower local quadratic loss after the same number of steps.
The existing optimizers for deep neural networks (DNNs) typically rely on either the $\ell_2$ norm or the $\ell_\infty$ norm, resulting in optimizers that do not adapt well to substantial changes in curvature across parameter dimensions. Generally, the training process of DNNs often exhibits strong curvature anisotropy in the early period, whereas in the later period, the training process of DNNs tends to move toward flatter regions with weaker anisotropy. Particularly, optimizers based on the \(\ell_2\)-norm are usually dominated by high-curvature directions, restricting updates of optimizers along with lower curvature direction and thus leading to a slower convergence rate. While optimizers based on the \(\ell_\infty\)-norm are prone to oscillations in flatter regions, due to the coordinate-wise updates of the same magnitude. To address these two extreme cases generated by $\ell_2$ and $\ell_\infty$ norms, we propose a novel $\ell_p$-norm scheme with a dynamical value of $p$ and incorporate it into stochastic gradient descent (SGD) and SGD with momentum (SGDM), leading to two novel optimizers with better generalization performance: ${\ell_p}$-SGD (LPSGD) and ${\ell_p}$-SGDM (LPSGDM). Particularly, the resulting optimizers suppress the dominance of high-curvature directions in the early period by utilizing a large $p$ ($p>2$), followed by a gradual decrease of $p$ toward 2 to enable more stable and refined updates, where the latter process is motivated by the cosine annealing strategy. We establish theoretical guarantees of the resulting algorithms and analyze that both LPSGD and LPSGDM achieve an \(O(T^{-1/2})\) convergence rate for the nonconvex setting. Extensive experiments are conducted on benchmark datasets, including CIFAR-10, CIFAR-100, and ImageNet-1K, with multiple DNNs such as VGG-11, ResNet-18, and ResNet-50.