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Posterior replicas and conditional information in Gaussian regression

OpenAI · September 27, 2026

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Abstract

For a signal with density bounded by L relative to uniform probability on $S^{d-1}$, we bound the information in a finite message W formed from exact Gaussian measurements, conditional on an independent projection revealed only to the analyst. For explicit row counts proportional to d, the bound is $O(H(W)/d+d+\log(2+\log L))$. Consequently, a finite-state learner with $o(d^2)$ persistent bits and a deterministic sample horizon needs $\Omega(d\log(1/\epsilon))$ fresh noiseless Gaussian measurements for constant-probability angular accuracy $0\lt \epsilon\le1/10$ under the uniform spherical prior.

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Cite

@misc{OAI:Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026,
  author = {{OpenAI}},
  title = {{Posterior replicas and conditional information in Gaussian regression}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf}{OAI:Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/140.md. The Lean statements have not been checked by people against the paper's claims.

Family 140: Memory–sample lower bounds for noiseless Gaussian regression

For fixed A > 0, a one-pass learner with $Ad^2$ persistent bits needs $\Omega_A(d\log(1/\epsilon))$ noiseless Gaussian samples to recover a unit vector to angular error $0\lt \epsilon\le1/10$ with probability 2/3, uniformly in accuracy for large d. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.

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