Result 139, Theoretical computer science

Subpolynomial query complexity for log-concave sampling

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.

Lean formalization Algorithm or complexity result

The bigger picture

Why it matters

Drawing random points from a log-concave distribution is guided by a convex potential, a function describing its shape. The manuscript reports that, under specific curvature assumptions, sampling needs surprisingly few information requests even in high dimensions.

What changes?

Consider twice continuously differentiable potentials V in d dimensions, with a supplied minimizer and curvature in every direction between 1 and 2. The target density is proportional to the exponential of minus V. For every fixed positive epsilon, the reported algorithm uses at most C_epsilon times d to the power epsilon exact value-and-gradient queries. C_epsilon depends on epsilon. This cap holds on every run, allowing unrestricted computation between queries. The output distribution has total-variation error at most 1/10.

What does that help mathematicians do?

The error guarantee limits the discrepancy in probability of any event to 1/10. Together with a reported logarithmic query lower bound for arbitrary randomized adaptive algorithms, the upper bound pins down the optimal power-law exponent in dimension as zero. Researchers can thus rule out any necessary positive-power query cost in this model, while also ruling out a query bound independent of dimension.

Are there practical applications?

The immediate value is foundational for sampling algorithms: it separates the amount of information needed about a potential from the computational work of producing a sample. Because queries are exact and computation between them is unrestricted, this is not a running-time guarantee. Practical efficiency would require additional control of computation and numerical precision.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Subpolynomial query complexity for well-conditioned log-concave sampling

September 26, 2026 55 pages Main result formalized in Lean

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 ℝ^d with a known minimizer and Hessian between I_d 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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026,
  author = {{OpenAI}},
  title = {{Subpolynomial query complexity for well-conditioned log-concave sampling}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026/article.pdf}{OAI:Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/139.md.

Subpolynomial query complexity for log-concave sampling

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Subpolynomial query complexity for well-conditioned log-concave sampling

Scope

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.

Comparator links

Result	Comparator statement
Subpolynomial query complexity for log-concave sampling	LogConcaveQuery.lean