The Gibbs Sampler is a general method for sampling high-dimensional distributions, dating back to 1971 [14]. In each step, we pick a random coordinate and re-sample that coordinate from the distribution induced by fixing all other coordinates. While it has become widely used over the past half-century, guarantees of efficient convergence have been elusive. Here we show that for convex bodies in R n with diameter D, the resulting Coordinate Hit-and-Run (CHAR) algorithm mixes in poly(n, D) steps. This is the first polynomial guarantee for this widely-used algorithm. We also give a lower bound on the mixing rate, showing that it is strictly worse than hit-and-run or the ball walk in the worst case.
We show that the the volume of a convex body in R n in the general membership oracle model can be computed with O(n 3 ψ 2 /ε 2 ) oracle queries, where ψ is the KLS constant 1 . With the current bound of ψ n 1 4 , this gives an O(n 3.5 /ε 2 ) algorithm, the first general improvement on the Lovász-Vempala O(n 4 /ε 2 ) algorithm from 2003. The main new ingredient is an O(n 3 ψ 2 ) algorithm for isotropic transformation, following which we can apply the O(n 3 /ε 2 ) volume algorithm of Cousins and Vempala for well-rounded convex bodies. A positive resolution of the KLS conjecture would imply an O(n 3 / 2 ) volume algorithm. We also give an efficient implementation of the new algorithm for convex polytopes defined by m inequalities in R n : polytope volume can be estimated in time O(mn c /ε 2 ) where c < 3.7 depends on the current matrix multiplication exponent and improves on the the previous best bound.
We study a unified approach and algorithm for constructive discrepancy minimization based on a stochastic process. By varying the parameters of the process, one can recover various state-of-the-art results. We demonstrate the flexibility of the method by deriving a discrepancy bound for smoothed instances, which interpolates between known bounds for worst-case and random instances.
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