Transformers as Measure-Theoretic Associative Memory: A Statistical Perspective and Minimax Optimality
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910008642371584 |
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| author | Kawata, Ryotaro Suzuki, Taiji |
| author_facet | Kawata, Ryotaro Suzuki, Taiji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01863 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Transformers as Measure-Theoretic Associative Memory: A Statistical Perspective and Minimax Optimality Kawata, Ryotaro Suzuki, Taiji Machine Learning 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. |
| title | Transformers as Measure-Theoretic Associative Memory: A Statistical Perspective and Minimax Optimality |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2602.01863 |