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CS-173 · computer science · mathematics · Computer science

Zero dimension exponent for well-conditioned sampling queries

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For $V\in C^{2}(R^d)$ with $V(0)=0$, $\nabla V(0)=0$ and $I\leq \nabla ^{2}V(x)\leq 2I$ everywhere, can its Gibbs law be sampled to total-variation error $\leq 1/10$ using at most $C_\varepsilon d^\varepsilon $ exact $(V,\nabla V)$ oracle queries on every execution, for each fixed $\varepsilon >0$, with unrestricted real computation between queries? Is the optimal query budget also at least c log d?

Short English statement

For $V\in C^{2}(R^d)$ with $V(0)=0$, $\nabla V(0)=0$ and $I\leq \nabla ^{2}V(x)\leq 2I$ everywhere, can its Gibbs law be sampled to total-variation error $\leq 1/10$ using at most $C_\varepsilon d^\varepsilon $ exact $(V,\nabla V)$ oracle queries on every execution, for each fixed $\varepsilon >0$, with unrestricted real computation between queries? Is the optimal query budget also at least c log d?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-173-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-173

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 139; 2026-09-26).

Zero dimension exponent for well-conditioned sampling queries

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 139; 2026-09-26), the asserted result is that exact value-and-gradient sampling for known-minimizer potentials with Hessian between $I$ and $2I$ has worst-case query budget at most $C_\varepsilon d^\varepsilon$, with a logarithmic lower bound and unrestricted internal computation.

Mathematical statement of the claimed result

Manuscript 1.
For $d\ge2$, let $V\in C^2(\mathbb R^d)$ satisfy $V(0)=0$, $\nabla V(0)=0$ and $I_d\preceq\nabla^2V(x)\preceq2I_d$ everywhere. The target is $\pi_V(dx)\propto e^{-V(x)}dx$. Exact adaptive first-order queries return $(V(x),\nabla V(x))$; algorithms are measurable and have unrestricted real computation and randomness between queries. Let $Q(d)$ be the minimum worst-execution query budget for total-variation error at most $1/10$ uniformly over this class. The claim gives \[ \forall\varepsilon>0\ \exists C_{\varepsilon}:\quad Q(d)\le C_{\varepsilon} d^{\varepsilon}, \qquad Q(d)\ge c\log d\quad(d\ge d_0) \] for absolute $c,d_0$. Thus the optimal dimension exponent is zero. These are oracle-query bounds, without a polynomial bit-time conclusion.

Outline of the argument

For manuscript 1 (family 139; 2026-09-26), transport-derived skew dynamics keep a stationary Gaussian-observation flow whose momentum records a conditional mean. Material-derivative bounds and nested noisy-center evaluations control the query count; stability and deterministic caps convert the analytic construction to a sampling procedure.

Formal verification

For manuscript 1 (family 139; 2026-09-26), OpenAI lists Lean statements for this assertion: exact value-and-gradient sampling for known-minimizer potentials with Hessian between $I$ and $2I$ has worst-case query budget at most $C_\varepsilon d^\varepsilon$, with a logarithmic lower bound and unrestricted internal computation.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are LogConcaveQuery.lean.

Their named declarations include OAI.LogConcaveSampling.exact_source_main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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