Abstract
arXiv:2605.27769v1 Announce Type: cross Abstract: We study the query complexity of sampling from high-dimensional Gaussian distributions using gradient information. In the standard oracle model, exact gradients expose only matrix-vector products with the precision matrix, leading to polynomial approximation barriers and a characteristic \(\sqrt{\kappa}\) dependence on the condition number. We show that this barrier disappears when the sampler is allowed to query \emph{smoothed scores}, namely gradients of the logarithms of the Gaussian-convolved densities. For a Gaussian target with precision matrix \(\Lambda\), a smoothed-score query at noise level \(\tau\) gives access to the resolvent \((\Lambda+\tau^{-1}I)^{-1}\). Combining geometrically spaced noise levels with sinc-quadrature rational approximation, we obtain a sampler with $q=O\!\left(\bigl(\log\kappa+\log(e\sqrt d/\delta_{\rm TV})\bigr)\log(e\sqrt d/\delta_{\rm TV})\right)$ smoothed-score queries for total variation error \(\delta_{\rm TV}\), improving the condition-number dependence from \(\sqrt{\kappa}\) to logarithmic. We also study finite-bit gradient oracles. Using coordinatewise quantization of the transformed smoothed-score answers and a final dithering step, we obtain a sampling scheme whose total communicated gradient information is polylogarithmic in \(\kappa\); in particular, for fixed dimension and accuracy, the bit complexity is \(O(\log^2\kappa)\). To complement these upper bounds, we introduce a channel-synthesis, or reverse-Shannon, converse technique for sampling lower bounds. This converts total-variation simulation guarantees into communication requirements and yields an \(\Omega(\log\kappa)\) lower bound on the required gradient information. Together, these results identify smoothed scores as a provably more informative oracle for sampling and give nearly matching upper and lower bounds for its finite-bit complexity.