Relevant and Irrelevant: A Renormalization Group Analysis of Transformer Attention
Using the language of Wilsonian renormalization group theory (RG), we treat the Transformer's attention mechanism as a perturbation of the trained MLP residual-stack fixed point and ask whether it constitutes a relevant, marginal, or irrelevant operator. We derive a fixed-point shift formula and obtain four testable predictions for the fixed-point geometry, effective rank profile, layer specificity, and perturbation decay spectrum. Testing these on synthetic Markov chain sequences with controlled correlation length, we find: (1) For large chains(long correlation), attention is strongly relevant: it closes a residual loss gap the MLP cannot bridge and drives a phase transition in representation space, with effective rank jumping above input dimensionality at layer 1 and stabilizing at a high-dimensional plateau. (2) For short chains(short correlation), attention is irrelevant: the Transformer converges to the same loss and fixed-point geometry as the MLP, though it contracts perturbations faster. (3) The transition is dominated by the first-layer head (L0H0), which accounts for more than 4 times the representational shift of any subsequent head, consistent with the prediction that the relevant operator acts before the MLP begins integrating out positional variation. (4) Perturbation decay experiments reveal a regime reversal: in the long correlation regime the Transformer selectively preserves slow Markov modes (5.4 times the dynamic range in decay length vs. 1.3 times for the MLP); in the short correlation regime it suppresses all modes faster than the MLP, with no spectral selectivity. Together, these results show that the relevance of attention is not a property of the architecture but of the spectral structure of the data-generating process, and that a first-order RG perturbation framework provides a predictive account of that difference.