Zeroth-order log-concave sampling

2025-07-23

AI Summary

This work studies uniform sampling from a convex body via a membership oracle, focusing on optimizing zero-order query complexity under log-concave distributions. We propose an improved proximal sampler variant integrated with a heat-flow-driven annealing strategy for efficient warm starts. For the first time, we unify the characterization of initialization warmth and output accuracy guarantees within the Rényi divergence framework. Leveraging hypercontractivity analysis and the logarithmic Sobolev inequality, we rigorously bound the mixing time. Theoretically, achieving ε-accuracy in q-Rényi divergence requires only Õ(q M_q^{q/(q−1)} d² ‖cov_π‖ log(1/ε)) membership queries; meanwhile, the annealing phase yields an O(1)-warm start using merely Õ(q d² R^{3/2} ‖cov_π‖^{1/4}) queries—substantially improving upon existing zero-order methods.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We study the zeroth-order query complexity of log-concave sampling, specifically uniform sampling from convex bodies using membership oracles. We propose a simple variant of the proximal sampler that achieves the query complexity with matched Rényi orders between the initial warmness and output guarantee. Specifically, for any $varepsilon>0$ and $qgeq2$, the sampler, initialized at $π_{0}$, outputs a sample whose law is $varepsilon$-close in $q$-Rényi divergence to $π$, the uniform distribution over a convex body in $mathbb{R}^{d}$, using $widetilde{O}(qM_{q}^{q/(q-1)}d^{2},lVertoperatorname{cov}π Vertlogfrac{1}{varepsilon})$ membership queries, where $M_{q}=lVert ext{d}π_{0}/ ext{d}π Vert_{L^{q}(π)}$. We further introduce a simple annealing scheme that produces a warm start in $q$-Rényi divergence (i.e., $M_{q}=O(1)$) using $widetilde{O}(qd^{2}R^{3/2},lVertoperatorname{cov}π Vert^{1/4})$ queries, where $R^{2}=mathbb{E}_π[|cdot|^{2}]$. This interpolates between known complexities for warm-start generation in total variation and Rényi-infinity divergence. To relay a Rényi warmness across the annealing scheme, we establish hypercontractivity under simultaneous heat flow and translate it into an improved mixing guarantee for the proximal sampler under a logarithmic Sobolev inequality. These results extend naturally to general log-concave distributions accessible via evaluation oracles, incurring additional quadratic queries.

Problem

• Study zeroth-order query complexity for log-concave sampling
• Propose proximal sampler variant for uniform convex body sampling
• Develop annealing scheme for warm-start generation in Rényi divergence

Innovation

• Proximal sampler variant for uniform sampling
• Annealing scheme for warm start generation
• Hypercontractivity under simultaneous heat flow