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Subpolynomial queries for log-concave sampling
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Subpolynomial queries for log-concave sampling
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| 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. |
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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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