Sharpness-Aware Curvature Learning

• The paper introduces a method that adjusts curvature in hyperbolic neural networks to produce flatter loss landscapes and boost out-of-sample accuracy.
• It employs a bilevel optimization framework with implicit differentiation to jointly fine-tune network weights and curvature parameters for enhanced stability.
• Empirical results demonstrate significant performance gains in classification, long-tailed, noisy, and few-shot learning scenarios, confirming practical robustness.

A sharpness-aware curvature learning method refers to a class of techniques that seek to optimize deep neural networks by explicitly accounting for the curvature of the loss landscape, targeting flatter solutions that empirically and theoretically correlate with improved generalization. By adapting or learning curvature parameters during training, these methods manage both the geometry of the embedding space (such as in hyperbolic neural networks) and the optimization dynamics, leading to quantitatively smoother loss surfaces and superior out-of-sample accuracy.

1. Geometric Foundations and Hyperbolic Neural Networks

Sharpness-aware curvature learning in hyperbolic neural networks (HNNs) is deeply rooted in the geometric properties of hyperbolic spaces, which are characterized by constant negative curvature. Early studies pioneered by Nickel and Kiela established the utility of the Poincaré ball model and associated optimization methods for representing hierarchical data structures (Fan et al., 24 Aug 2025).

Hyperbolic geometry enables exponentially increasing capacity with radius, which naturally encodes tree-like or graph data with minimal distortion. In HNNs, the choice of curvature governs not only the separation between embeddings but also the local smoothness and optimization trajectory. Inappropriate curvature selection can cause the network to converge to suboptimal solutions, underscoring the necessity to learn or adapt curvature during training.

2. Curvature-Sharpness Interaction and Loss Landscape Smoothness

Sharpness—typically measured as the largest eigenvalue of the loss Hessian matrix or related metrics—quantifies how rapidly the loss increases in the vicinity of a model's parameters. Sharp minima (high curvature) are known to generalize poorly due to their sensitivity to small perturbations, while flat minima (low curvature) are preferred for generalization.

The sharpness-aware curvature learning method proposed for HNNs applies a “scope sharpness measure” that links curvature to the local smoothness of the loss landscape. By minimizing this measure, the method smooths the loss surface. This principle is consistent with PAC-Bayesian bounds derived for HNNs, explicitly showing that curvature learning impacts generalization error by regulating the smoothness of the parameter-to-output map (Fan et al., 24 Aug 2025).

3. Bilevel Optimization and Implicit Differentiation

To learn appropriate curvature parameters, the method formulates a bilevel optimization scheme. The lower-level problem consists of the standard HNN optimization for a given curvature, while the upper-level problem seeks to minimize a generalization objective—typically, a surrogate for loss landscape sharpness or the generalization gap.

Gradient-based bilevel optimization is challenging, especially when curvature parameters reside on Riemannian manifolds. The proposed method employs implicit differentiation to efficiently estimate gradients of curvature with respect to generalization objectives. The approximation error incurred by implicit differentiation is shown to be upper-bounded, and convergence guarantees are established through bounding the gradient norms of HNNs (Fan et al., 24 Aug 2025).

4. Empirical Results in Classification, Long-Tailed Data, Noisy Data, and Few-Shot Learning

Experimental validation covers four challenging areas:

• Standard classification tasks
• Long-tailed data distributions
• Noisy label regimes
• Few-shot learning scenarios

In all cases, HNNs equipped with sharpness-aware curvature learning outperform conventional (fixed-curvature) HNNs. The method demonstrates notable gains in generalization accuracy, especially in settings where the underlying data geometry is naturally hierarchical or where stability under noise and limited data is critical.

5. Context within Broader Research and Related Work

Seminal contributions from Ungar and others have established analytic frameworks for Riemannian optimization and curvature learning in non-Euclidean spaces. More recent efforts by Cìgneul, Ganea, Hofmann, and others have extended these frameworks to deep learning architectures, investigating the interplay between curvature, optimization, and representation quality (Fan et al., 24 Aug 2025). Related methods in the literature include sharpness-aware minimization (SAM) and its extensions, which inspired adaptations of curvature-aware optimization to hyperbolic settings. The present method builds on these insights by integrating sharpness criteria and curvature learning via bilevel optimization and implicit gradient computation.

6. Connections to Generalization Theory and Data Geometry

PAC-Bayesian generalization bounds for HNNs derived in this work reveal that curvature directly modulates data-dependent smoothness and hence generalizability. This connection underlines the importance of adaptive curvature in managing both the expressivity (capacity to represent hierarchical and complex structures) and the stability of learned representations. Additional evidence from learning with noisy and long-tailed labels suggests that aligning network geometry with intrinsic data geometry produces quantifiable gains in robustness and effective sample efficiency (Fan et al., 24 Aug 2025).

7. Open Problems and Future Directions

Sharpness-aware curvature learning is positioned at the intersection of geometric deep learning, PAC-Bayesian generalization analysis, and advanced optimization on Riemannian manifolds. Important questions remain:

• How best to scale implicit gradient estimation for very large architectures.
• How to integrate sharpness-aware curvature control with other regularization mechanisms.
• What are the optimal representations for non-tree-like data in variable-curvature geometries?
• When and how does curvature adaptation interact with stochastic optimization and noise robustness?

The methodologies and theoretical advances described in the referenced work (Fan et al., 24 Aug 2025) motivate further exploration into curvature-adaptive learning across a broad spectrum of structured data, complex tasks, and architectural modalities.