Research on transformer expressivity shows whether a transformer is capable of solving a given task, but gives little indication of whether the solution, if learned, is generalizable to longer input lengths.
We study this question through normalized exact-solution volume (NESV): the fraction of a bounded parameter region that achieves an exact solution on every input of length $n$.
Asymptotic Bounds on NESV
For fixed-width, single-layer transformers with $\log n$-scaled attention, we establish asymptotic bounds on NESV for four tasks: FIRST ($Θ(1)$), MAJORITY ($Θ(1/(n\log n))$), INDEX ($Θ(1/n^3)$), and PARITY ($0$).
These results are consistent with previous empirical results: the faster the exact-solution volume decays with input length, the harder it is to length-generalize on that task.
INDEX Analysis
Looking deeper into INDEX, our volume analysis reveals two error sources that grow with $n$.
Consequently, we study a transformer model that would structurally eliminate one of the terms, theoretically improving the NESV bound to $Θ(n^{-1})$, and empirically achieving 85% accuracy when tested at $10\times$ the training length, compared with the 60% accuracy of the original model.
We conclude that volume analysis may be a useful approach to identify concrete sources of length sensitivity and thus provide insights into task-specific model refinements.