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Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression

OpenAI · September 27, 2026

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Abstract

Let a finite-state streaming learner estimate a uniformly random unit vector from independent exact Gaussian linear measurements. We prove that $o(d^2)$ bits of persistent memory and angular success probability at least 2/3 require at least $2^{-16}d\log_2(1/\epsilon)$ samples for all sufficiently large d, uniformly for $0\lt \epsilon\le1/10$. The proof conditions each batch on its observed projection and controls the resulting random residual subsphere.

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Cite

@misc{OAI:Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026,
  author = {{OpenAI}},
  title = {{Subsphere methods for memory--sample lower bounds in noiseless Gaussian regression}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf}{OAI:Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-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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