A Unified Framework for Critical Scaling of Inverse Temperature in Self-Attention
Provides a theoretical foundation for stabilizing long-context attention in transformers, addressing a key bottleneck in scaling to longer sequences.
The paper unifies conflicting inverse-temperature scaling laws for self-attention by introducing a gap-counting function that determines the critical scale for softmax concentration, resolving discrepancies between prior results ranging from (log n)^{1/2} to (log n)^2.
Length-dependent logit rescaling is widely used to stabilize long-context self-attention, but existing analyses and methods suggest conflicting inverse-temperature laws for the context length $n$, ranging from $(\log n)^{1/2}$ to $\log n$ and $(\log n)^2$. We provide a general theory showing that the desirable scale is determined by the gap-counting function $N_n$ of each attention row. Counting how many competitors lie within each gap from the maximum, we define an upper-tail accumulation scale and prove that it gives the critical inverse-temperature scale for softmax concentration: below this scale, the top competitors remain unseparated, whereas above it, the attention entropy collapses. This framework unifies prior scaling laws as different $N_n$ and yields a direct diagnostic for attention-score families, from idealized theoretical models to more practical transformers.