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Oracle-based Uniform Sampling from Convex Bodies

Thanh Dang, Jiaming Liang

TL;DR

This work develops unbiased Markov chain Monte Carlo methods to sample the uniform distribution on a convex body $K$ by embedding it into the Alternating Sampling Framework (ASF)/proximal sampler and implementing the Restricted Gaussian Oracle (RGO) via rejection sampling. It provides two concrete RGO schemes—one using a projection oracle and one using a separation oracle—each yielding non-asymptotic guarantees in Rényi divergence $\mathcal{R}_q$ and $\chi^2$-divergence, and avoids restart failures common in prior approaches. The authors derive explicit iteration and rejection-cost bounds that scale with the dimension and isoperimetric constants, showing competitive performance with Ball and Hit-and-Run-type samplers under warm-start assumptions. They also discuss extension to cutting-plane–based RGO and outline directions for broader log-concave sampling on $K$, highlighting practical impact for volume computation and Bayesian truncated models.

Abstract

We propose new Markov chain Monte Carlo algorithms to sample a uniform distribution on a convex body $K$. Our algorithms are based on the Alternating Sampling Framework/proximal sampler, which uses Gibbs sampling on an augmented distribution and assumes access to the so-called restricted Gaussian oracle (RGO). The key contribution of this work is the efficient implementation of RGO for uniform sampling on $K$ via rejection sampling and access to either a projection oracle or a separation oracle on $K$. In both oracle cases, we establish non-asymptotic complexities to obtain unbiased samples where the accuracy is measured in Rényi divergence or $χ^2$-divergence.

Oracle-based Uniform Sampling from Convex Bodies

TL;DR

This work develops unbiased Markov chain Monte Carlo methods to sample the uniform distribution on a convex body by embedding it into the Alternating Sampling Framework (ASF)/proximal sampler and implementing the Restricted Gaussian Oracle (RGO) via rejection sampling. It provides two concrete RGO schemes—one using a projection oracle and one using a separation oracle—each yielding non-asymptotic guarantees in Rényi divergence and -divergence, and avoids restart failures common in prior approaches. The authors derive explicit iteration and rejection-cost bounds that scale with the dimension and isoperimetric constants, showing competitive performance with Ball and Hit-and-Run-type samplers under warm-start assumptions. They also discuss extension to cutting-plane–based RGO and outline directions for broader log-concave sampling on , highlighting practical impact for volume computation and Bayesian truncated models.

Abstract

We propose new Markov chain Monte Carlo algorithms to sample a uniform distribution on a convex body . Our algorithms are based on the Alternating Sampling Framework/proximal sampler, which uses Gibbs sampling on an augmented distribution and assumes access to the so-called restricted Gaussian oracle (RGO). The key contribution of this work is the efficient implementation of RGO for uniform sampling on via rejection sampling and access to either a projection oracle or a separation oracle on . In both oracle cases, we establish non-asymptotic complexities to obtain unbiased samples where the accuracy is measured in Rényi divergence or -divergence.

Paper Structure

This paper contains 14 sections, 16 theorems, 73 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation, definitions, and assumptions
  4. Alternating sampling framework
  5. Projection oracle-based proximal sampling
  6. Separation oracle-based proximal sampling
  7. Concluding remarks
  8. Technical lemmas and proofs
  9. Supplementary materials for Section \ref{['sec:separation_uniform']}
  10. Results about the Cutting Plane method by jiang2020cuttingplane
  11. Proof of Lemma \ref{['lem:comparealltheP']}
  12. Proof of Proposition \ref{['prop:averagerejection_separation']}
  13. Further explanation for Algorithm \ref{['alg:separation:Psamplingforseparation']}
  14. Algorithms for uniform sampling on convex bodies

Key Result

Lemma 2.1

kook2024inandout Let $\pi$ be the uniform distribution over $K$ and $K\subset \mathbb{R}^d$ be a convex body with diameter $D$, where $D=\max_{x,y\in K}\left\lVert x-y \right\rVert$. Then we have $C_{\mathrm{PI}}(\pi)= \mathcal{O}\left( \left\lVert \mathrm{Cov}{\left( \pi\right)} \right\rVert_{\ope

Theorems & Definitions (26)

  • Lemma 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Remark 2.4
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 16 more