Oracle-based Uniform Sampling from Convex Bodies

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.