Hyperbolic graph representation learning has emerged as a powerful paradigm for modeling hierarchical data. Unlike Euclidean space, which exhibits polynomial volume growth, hyperbolic space grows exponentially with radius, naturally accommodating tree-like structures without the geometric distortion known as Euclidean crowding. Despite strong empirical performance, the theoretical mechanisms underlying hyperbolic embeddings remain poorly understood. In this work, we provide a geometric quantifcation research for Euclidean crowding, a generalization-theoretic explanation for the superiority of hyperbolic space on hierarchical data, and further develop an adaptive Euclidean-hyperbolic fusion learning framework toward bridging this gap. Specifically, we \textbf{first} establish a geometric quantification of Euclidean crowding by analyzing layer-wise minimum separation in node embeddings, proving that fixed-dimensional Euclidean embeddings of tree-like graphs inevitably exhibit layer-wise crowding that leads to weaker margin-based generalization bounds. \textbf{Second}, we provide a generalization-bound for the superiority of hyperbolic geometry on hierarchical data. \textbf{Third}, we propose EH-Fusion, a geometry-adaptive framework that lifts Euclidean GNN representations into hyperbolic space via exponential mapping and adaptively fuses both geometries based on node-level neighborhood expansion patterns. Extensive experiments validate our theoretical insights and demonstrate the effectiveness of EH-Fusion on benchmark datasets.
Xiangchi Yuan, Xingjian Diao, Zheyuan Liu, Chunhui Zhang, Dachuan Shi, Kejing Xia, Zhenbang Du, Shenglong Yao 等 10 位作者
LLM post-training often optimizes multiple objectives with unequal priorities, such as correctness and reasoning efficiency. However, the widely used additive reward aggregation in RL can violate these priorities: when primary and auxiliary advantages conflict, a sufficiently large auxiliary signal can overturn the preference induced by the primary objective. We term this problem \emph{advantage priority inversion}. Although reward-level gating can protect priority more directly, it still removes potentially useful auxiliary distinctions among rollout samples that fail the primary condition. To address this trade-off, we propose Primary Advantage Policy Optimization (\textbf{PAPO}), a priority-aware advantage aggregation method that asymmetrically combines a primary and an auxiliary advantage. PAPO retains auxiliary supervision when the objectives agree and smoothly attenuates conflicting auxiliary signals, preserving the primary sign during fusion for \emph{primary-consistent ordering}. This rule applies to sequence-level advantages in GRPO and token-level advantages in actor--critic PPO. Experiments on multiple domains and models show that PAPO consistently achieves the best primary objective while maintaining best or competitive auxiliary-objective performance across both GRPO- and PPO-based optimization. \href{https://xiangchi-yuan.github.io/papo-project/}{Code is available}.