Smoothed Score Queries and the Complexity of Sampling

2026-05-26

AI Summary

This work addresses the challenge of high-dimensional Gaussian sampling, where conventional gradient-based oracles—providing only matrix-vector products with the precision matrix—lead to a sampling complexity that scales as √κ with the condition number κ. To overcome this limitation, the paper introduces a novel oracle based on smoothed scores, defined as the gradient of the log-density of a Gaussian-convolved distribution. By integrating geometrically spaced noise levels, sinc-quadrature rational approximation, coordinate quantization, and dithered post-processing, the proposed method eliminates the √κ dependence for the first time, reducing the condition-number dependence to logarithmic. Under total variation error δ_TV, the query complexity is O((log κ + log(e√d/δ_TV))·log(e√d/δ_TV)), and for fixed dimension and accuracy, the bit complexity achieves O(log²κ). An information-theoretic lower bound of Ω(log κ) is also established, demonstrating near-tightness.

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 \(\sqrtκ\) 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 \(Λ\), a smoothed-score query at noise level \(τ\) gives access to the resolvent \((Λ+τ^{-1}I)^{-1}\). Combining geometrically spaced noise levels with sinc-quadrature rational approximation, we obtain a sampler with $q=O\!\left(\bigl(\logκ+\log(e\sqrt d/δ_{\rm TV})\bigr)\log(e\sqrt d/δ_{\rm TV})\right)$ smoothed-score queries for total variation error \(δ_{\rm TV}\), improving the condition-number dependence from \(\sqrtκ\) 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 \(κ\); in particular, for fixed dimension and accuracy, the bit complexity is \(O(\log^2κ)\). 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 \(Ω(\logκ)\) 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.

Problem

• sampling
• condition number
• smoothed scores
• query complexity
• gradient oracle

Innovation

• smoothed score queries
• sampling complexity
• condition number dependence
• finite-bit communication
• channel synthesis lower bound