Transformers as Measure-Theoretic Associative Memory: A Statistical Perspective and Minimax Optimality

Transformers excel through content-addressable retrieval and the ability to exploit contexts of, in principle, unbounded length. We recast associative memory at the level of probability measures, treating a context as a distribution over tokens and viewing attention as an integral operator on measures. Concretely, for mixture contexts $ν= I^{-1} \sum_{i=1}^I μ^{(i^*)}$ and a query $x_{\mathrm{q}}(i^*)$, the task decomposes into (i) recall of the relevant component $μ^{(i^*)}$ and (ii) prediction from $(μ_{i^*},x_\mathrm{q})$. We study learned softmax attention (not a frozen kernel) trained by empirical risk minimization and show that a shallow measure-theoretic Transformer composed with an MLP learns the recall-and-predict map under a spectral assumption on the input densities. We further establish a matching minimax lower bound with the same rate exponent (up to multiplicative constants), proving sharpness of the convergence order. The framework offers a principled recipe for designing and analyzing Transformers that recall from arbitrarily long, distributional contexts with provable generalization guarantees.