In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies

We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in Rényi divergence (which implies TV, ${\mathcal{W}}_2$, KL, ${\chi }^2$). The proof departs from known approaches for polytime algorithms for the problem – we utilize a stochastic diffusion perspective to show contraction to the target distribution with the rate of convergence determined by functional isoperimetric constants of the target distribution.