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arXiv · 2512.13728

CurvaDion: Curvature-Adaptive Distributed Orthonormalization

Bhavesh Kumar·Roger Jin·Jeffrey Quesnelle
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Abstract

As language models scale to trillions of parameters, distributed training across many GPUs becomes essential, yet gradient synchronization over high-bandwidth, low-latency networks remains a critical bottleneck. While recent methods like Dion reduce per-step communication through low-rank updates, they synchronize at every step regardless of the optimization landscape. We observe that synchronization requirements vary dramatically throughout training: workers naturally compute similar gradients in flat regions, making frequent synchronization redundant, while high-curvature regions require coordination to prevent divergence. We introduce CurvaDion, which uses Relative Maximum Momentum Change (RMMC) to detect high-curvature regions requiring synchronization. RMMC leverages momentum dynamics which are already computed during optimization as a computationally tractable proxy for directional curvature, adding only $\mathcal{O}(d)$ operations per layer. We establish theoretical connections between RMMC and loss curvature and demonstrate that CurvaDion achieves 99\% communication reduction while matching baseline convergence across models from 160M to 1.3B parameters.

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Bhavesh Kumar, Roger Jin, Jeffrey Quesnelle. 2025-12-13. CurvaDion: Curvature-Adaptive Distributed Orthonormalization. https://arxiv.org/abs/2512.13728

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