Non-convex Uniform Sampling Bounds for Ball and Speedy Walks

Determine whether the Ball walk or the Speedy walk admit a bound on the number of proper steps for uniform sampling over general non-convex sets analogous to the bound established for the In-and-Out algorithm.

Background

The authors show that their diffusion-based In-and-Out algorithm yields a bound on the number of proper steps that holds for general non-convex bodies and any feasible start, deduced under an M-warm start. They contrast this with existing geometric random walks.

They explicitly remark that such bounds are not available for the classic Ball and Speedy walks in non-convex settings, highlighting a gap in current understanding of these algorithms beyond convex bodies.