Rank-Aware Spectral Bounds on Attention Logits for Stable Low-Precision Training

Seyed Morteza Emadi
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:28021-28039, 2026.

Abstract

Attention scores in transformers are bilinear forms $S_{ij} = x_i^\top M x_j / \sqrt{d_h}$ whose maximum magnitude governs overflow risk in low-precision training. We derive a rank-aware concentration inequality: when the interaction matrix $M = W^Q W^{K\top}$ has rank $r \ll d$, tail probabilities for $\max_{i,j}|S_{ij}|$ decay as $\exp(-d^{2}\alpha^{2}/(\gamma r))$ rather than $\exp(-d\alpha^{2})$, where $\gamma > 1$. For transformer attention where $r = d_h$, this yields $8$–$28\times$ tighter concentration than rank-agnostic bounds in modern architectures. We apply this result to FP8 training, deriving geometry-aware scale factors that provide principled overflow guarantees without observing activations. The method computes per-layer scales from the spectral norm $\|W^Q W^{K\top}\|_2$ via implicit power iteration, includes a grouped query attention formulation that avoids key expansion, and remains compatible with fused attention kernels. Across GPT-2 XL to Llama-2-70B, geometry-aware scaling eliminates overflows in transient scenarios where delayed scaling fails, while achieving comparable downstream MMLU accuracy.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-emadi26a, title = {Rank-Aware Spectral Bounds on Attention Logits for Stable Low-Precision Training}, author = {Emadi, Seyed Morteza}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {28021--28039}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/emadi26a/emadi26a.pdf}, url = {https://proceedings.mlr.press/v306/emadi26a.html}, abstract = {Attention scores in transformers are bilinear forms $S_{ij} = x_i^\top M x_j / \sqrt{d_h}$ whose maximum magnitude governs overflow risk in low-precision training. We derive a rank-aware concentration inequality: when the interaction matrix $M = W^Q W^{K\top}$ has rank $r \ll d$, tail probabilities for $\max_{i,j}|S_{ij}|$ decay as $\exp(-d^{2}\alpha^{2}/(\gamma r))$ rather than $\exp(-d\alpha^{2})$, where $\gamma > 1$. For transformer attention where $r = d_h$, this yields $8$–$28\times$ tighter concentration than rank-agnostic bounds in modern architectures. We apply this result to FP8 training, deriving geometry-aware scale factors that provide principled overflow guarantees without observing activations. The method computes per-layer scales from the spectral norm $\|W^Q W^{K\top}\|_2$ via implicit power iteration, includes a grouped query attention formulation that avoids key expansion, and remains compatible with fused attention kernels. Across GPT-2 XL to Llama-2-70B, geometry-aware scaling eliminates overflows in transient scenarios where delayed scaling fails, while achieving comparable downstream MMLU accuracy.} }
Endnote
%0 Conference Paper %T Rank-Aware Spectral Bounds on Attention Logits for Stable Low-Precision Training %A Seyed Morteza Emadi %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-emadi26a %I PMLR %P 28021--28039 %U https://proceedings.mlr.press/v306/emadi26a.html %V 306 %X Attention scores in transformers are bilinear forms $S_{ij} = x_i^\top M x_j / \sqrt{d_h}$ whose maximum magnitude governs overflow risk in low-precision training. We derive a rank-aware concentration inequality: when the interaction matrix $M = W^Q W^{K\top}$ has rank $r \ll d$, tail probabilities for $\max_{i,j}|S_{ij}|$ decay as $\exp(-d^{2}\alpha^{2}/(\gamma r))$ rather than $\exp(-d\alpha^{2})$, where $\gamma > 1$. For transformer attention where $r = d_h$, this yields $8$–$28\times$ tighter concentration than rank-agnostic bounds in modern architectures. We apply this result to FP8 training, deriving geometry-aware scale factors that provide principled overflow guarantees without observing activations. The method computes per-layer scales from the spectral norm $\|W^Q W^{K\top}\|_2$ via implicit power iteration, includes a grouped query attention formulation that avoids key expansion, and remains compatible with fused attention kernels. Across GPT-2 XL to Llama-2-70B, geometry-aware scaling eliminates overflows in transient scenarios where delayed scaling fails, while achieving comparable downstream MMLU accuracy.
APA
Emadi, S.M.. (2026). Rank-Aware Spectral Bounds on Attention Logits for Stable Low-Precision Training. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:28021-28039 Available from https://proceedings.mlr.press/v306/emadi26a.html.

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