Subpolynomial query complexity for log-concave sampling
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For C2 potentials with a supplied minimizer and \(I\preceq\nabla^2V\preceq2I\), proves that sampling within total variation 1/10 requires only \(C_\varepsilon d^\varepsilon\) exact value-and-gradient queries for every fixed ε > 0. The bound holds on every run, with unrestricted computation between queries. A logarithmic lower bound also holds, so the optimal power-law exponent in this oracle model is zero. (OpenAI’s Lean note)
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In the oracle model above, for every \varepsilon>0 there is C_\varepsilon<\infty such that
For every fixed ε > 0, we give a sampling algorithm using at most \(C_\varepsilon d^\varepsilon\) exact first-order queries on every execution for C2 potentials on ℝd with a known minimizer and Hessian between Id and \(2I_d\). The output has total-variation distance at most 1/10 from the target Gibbs law. Computation between queries is unrestricted. We also prove an \(\Omega(\log d)\) query lower bound for arbitrary randomized adaptive algorithms, determining the optimal dimension exponent to be zero.
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Their note:None; the claim is about the number of exact value-and-gradient queries only (no time/bit-cost statement), as in the summary.
Definitions the group asks readers to compare with the paper:Hand-rolled oracle model (OracleAlgorithm, OracleAlgorithm.history/run, TVAtMost := ∀ s measurable, |μ s - ν s| ≤ ε, queryComplexity := sInf over feasible integer budgets in ℕ∞); non-uniform in d (one algorithm per (d, q))
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The formalized result determines the dimension exponent of exact value-and-gradient query complexity for well-conditioned log-concave sampling. For potentials with $V(0)=0$, $\nabla V(0)=0$, and $I\le\nabla^2V\le2I$, the least worst-case query budget achieving total-variation error at most $1/10$ is at most $C_\varepsilon d^\varepsilon$ for every $\varepsilon>0$, and at least $c\log d$ eventually. Hence the infimal polynomial exponent is zero. Algorithms may be measurable, randomized, and adaptive; arithmetic and bit costs are not bounded.
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Subpolynomial query complexity for log-concave samplingLogConcaveQuery
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for well-conditioned log-concave sampling, the query complexity is at most C_ε d^ε for every ε > 0 and at least c log d, so the polynomial exponent is zero.
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Subpolynomial query complexity for log-concave samplingLogConcaveQuery
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Date
8 October 2026
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at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.