John's Walk

We present an affine-invariant random walk for drawing uniform random samples from a convex body $\mathcal{K} \subset \mathbb{R}^n$ for which the maximum volume inscribed ellipsoid, known as John's ellipsoid, may be computed. We consider a polytope $\mathcal{P} = \{x \in \mathbb{R}^n \mid Ax \leq 1\}$ where $A \in \mathbb{R}^{m \times n}$ as a special case. Our algorithm makes steps using uniform sampling from the John's ellipsoid of the symmetrization of $\mathcal{K}$ at the current point. We show that from a warm start, the random walk mixes in $\widetilde{O}(n^7)$ steps where the log factors depend only on constants determined by the warm start and error parameters (and not on the dimension or number of constraints defining the body). This sampling algorithm thus offers improvement over the affine-invariant Dikin Walk for polytopes (which mixes in $\widetilde{O}(mn)$ steps from a warm start) for applications in which $m \gg n$. Furthermore, we describe an $\widetilde{O}(mn^{\omega+1} + n^{2\omega+2})$ algorithm for finding a suitably approximate John's ellipsoid for a symmetric polytope based on Vaidya's algorithm, and show the mixing time is retained using these approximate ellipsoids (where $\omega < 2.373$ is the current value of the fast matrix multiplication constant).