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A Simple Analysis of the Dikin Walk

Sampling points from the uniform distribution on a polytope is a well-studied problem, and is an important ingredient in several computational tasks involving polytopes, such as volume estimation. This is achieved by setting up a random walk inside the polytope, with its stationary distribution being uniform in the interior of the polytope. Kannan-Narayanan and Narayanan proposed the Dikin walk based on interior point methods, where the next point is sampled, roughly, from the Dikin ellipsoid at the current point. In this paper, we give a simple proof of the mixing time of the Dikin walk, using well-known properties of Gaussians, and concentration of Gaussian polynomials.