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arXiv · 2609.35392

The Hidden Ratio in Adam: Stable Structure, Compression, and Sign Dynamics

Yihe Zhou·Tongtian Zhu·Yingxiao Huo·Satya Prakash Dash·Can Wang·Samuel Kaski·Mingfei Sun
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Abstract

Adam is the default optimizer for training modern deep neural networks, yet its adaptive behavior remains poorly understood due to the complex interaction between its first- and second-moment exponential moving averages (EMAs). We study Adam in the tied-$β$ regime, where the two EMA decay rates are equal, and show that its adaptive dynamics can be expressed through a transformed ratio with approximately scale-stable behavior. Empirically, this transformed ratio exhibits a stable, heavy-tailed distribution across tasks, model scales, and training stages, in contrast to the variability of raw moment magnitudes. This empirical stability has both practical and conceptual consequences. First, we derive a recurrence for the transformed ratio, yielding a reparameterization of Adam that replaces the second moment with a compressible state. Leveraging its stable distribution, we show that a fixed 4-bit codebook is sufficient in our experiments to store this state without auxiliary scaling, achieving performance competitive with full-precision Adam. Second, the transformed ratio view clarifies Adam's connection to sign-based methods: Adam reduces to sign-based momentum modulated by the transformed ratio, and replacing it with a constant recovers Signum as a limiting case. This perspective further provides a simple rule for transferring learning rates between the two methods. Together, these results suggest that tied-$β$ Adam admits a simple and approximately stable ratio structure underlying its adaptive behavior and demonstrate its utility for both analysis and efficient implementation.

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Yihe Zhou, Tongtian Zhu, Yingxiao Huo, Satya Prakash Dash, Can Wang, Samuel Kaski, Mingfei Sun. 2026-09-28. The Hidden Ratio in Adam: Stable Structure, Compression, and Sign Dynamics. https://arxiv.org/abs/2609.35392

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