Geodesic Walks in Polytopes

We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from the interior of polytopes in $\mathbb{R}^{n}$ specified by $m$ inequalities. The walk is a discrete-time simulation of a stochastic differential equation on the Riemannian manifold equipped with the metric induced by the Hessian of a convex function; each step is the solution of an ordinary differential equation (ODE). The resulting sampling algorithm for polytopes mixes in $O^{*}(mn^{\frac{3}{4}})$ steps. This is the first walk that breaks the quadratic barrier for mixing in high dimension, improving on the previous best bound of $O^{*}(mn)$ by Kannan and Narayanan for the Dikin walk. We also show that each step of the geodesic walk (solving an ODE) can be implemented efficiently, thus improving the time complexity for sampling polytopes. Our analysis of the geodesic walk for general Hessian manifolds does not assume positive curvature and might be of independent interest.