A Rod Flow Model for Adam at the Edge of Stability
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913099449106432 |
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| author | Regis, Eric Chewi, Sinho |
| author_facet | Regis, Eric Chewi, Sinho |
| contents | Cohen et al. (arXiv:2207.14484) observed that adaptive gradient methods such as Adam operate at the edge of stability. While there has been significant work on continuous-time modeling of gradient descent at the edge of stability, extending these models to momentum methods remains underdeveloped. In the gradient descent setting, Regis et al. (arXiv:2602.01480) introduced rod flow, which models consecutive iterates as an extended one-dimensional object -- a "rod." Here we extend rod flow to Adam by working in the joint phase space of parameters and first moment $(w, m)$ and treating the second moment $ν$ as a smooth auxiliary variable. We also develop rod flows for heavy ball momentum, Nesterov momentum, and scalar and per-component versions of RMSProp, Adam, and NAdam. For all eight optimizers, we empirically evaluate rod flow on representative machine learning architectures, where it tracks the discrete iterates through the edge-of-stability regime significantly more accurately than the corresponding stable flow. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_06821 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Rod Flow Model for Adam at the Edge of Stability Regis, Eric Chewi, Sinho Machine Learning Artificial Intelligence Optimization and Control Cohen et al. (arXiv:2207.14484) observed that adaptive gradient methods such as Adam operate at the edge of stability. While there has been significant work on continuous-time modeling of gradient descent at the edge of stability, extending these models to momentum methods remains underdeveloped. In the gradient descent setting, Regis et al. (arXiv:2602.01480) introduced rod flow, which models consecutive iterates as an extended one-dimensional object -- a "rod." Here we extend rod flow to Adam by working in the joint phase space of parameters and first moment $(w, m)$ and treating the second moment $ν$ as a smooth auxiliary variable. We also develop rod flows for heavy ball momentum, Nesterov momentum, and scalar and per-component versions of RMSProp, Adam, and NAdam. For all eight optimizers, we empirically evaluate rod flow on representative machine learning architectures, where it tracks the discrete iterates through the edge-of-stability regime significantly more accurately than the corresponding stable flow. |
| title | A Rod Flow Model for Adam at the Edge of Stability |
| topic | Machine Learning Artificial Intelligence Optimization and Control |
| url | https://arxiv.org/abs/2605.06821 |