VC Dimension and Expressivity of Real-Valued Transformers

Whereas previous results on abilities and limitations of transformers have restricted the definition of transformers in various ways, here we study softmax-attention, multi-layer transformers operating on real values, with very few additional assumptions. Applying results from real geometry, we obtain upper bounds on the VC dimension and split VC dimension of such transformers ($O(n^4)$ and $O(n^6)$, respectively, where $n$ is the input length). Conversely, we also construct specific transformers witnessing lower bounds on these quantities ($\Omega (n)$ in each case). These results have some notable consequences. For example, within the class of symmetric (permutation-invariant) functions, we show that transformers can uniformly express all functions over an alphabet of one symbol and non-uniformly express all functions over an alphabet of two symbols, but cannot (even non-uniformly) express some functions over an alphabet of six symbols. We also prove limitations on how many bits of a real number a transformer can access.