For uniform keys on the d-dimensional sphere, softmax attention becomes selective at inverse temperature scaling β_n* ≍ n^{2/(d-1)}, with explicit limiting laws for attention weights and outputs in each regime.
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Kernel gradient flows attain minimax-optimal sup-norm generalization rates and admit simultaneous confidence bands with near-optimal widths under standard capacity-source conditions.
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Scaling Limits of Long-Context Transformers
For uniform keys on the d-dimensional sphere, softmax attention becomes selective at inverse temperature scaling β_n* ≍ n^{2/(d-1)}, with explicit limiting laws for attention weights and outputs in each regime.
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Optimal Confidence Band for Kernel Gradient Flow Estimator
Kernel gradient flows attain minimax-optimal sup-norm generalization rates and admit simultaneous confidence bands with near-optimal widths under standard capacity-source conditions.