Randomized interior point methods for sampling and optimization

Abstract

We present a Markov Chain, "Dikin walk", for sampling from a convex body equipped with a self-concordant barrier. This Markov Chain corresponds to a natural random walk with respect to a Riemannian metric defined using the Hessian of the barrier function.For every convex set of dimension n, there exists a self-concordant barrier whose self-concordance parameter is O(n). Consequently, a rapidly mixing Markov Chain of the kind we describe can be defined (but not always be efficiently implemented) on any convex set. We use these results to design an algorithm consisting of a single random walk for optimizing a linear function on a convex set. Using results of Barthe [2] and Bobkov and Houdré [5], on the isoperimetry of products of weighted Riemannian manifolds, we obtain sharper upper bounds on the mixing time of a Dikin walk on products of convex sets than the bounds obtained from a direct application of the Localization Lemma. The results in this paper generalize previous results of [12] from polytopes to spectrahedra and beyond, and improve upon those results in a special case when the convex set is a direct product of lower dimensional convex sets. This Markov Chain like the chain described in [12] is affine-invariant.MSC classification: 65C40, 90C30

The mixing time of the Dikin walk in a polytope—A simple proof

“…The uni-form Dikin walk, considered by [6], sampled the new point z from the uniform distribution in this ellipsoid. In the Gaussian Dikin walk, considered by [9], z is sampled from g x , a multivariate Gaussian distribution centered at x with covariance matrix r 2 n H(x) −1 , where r is a constant. Thus, the density of the distribution is given by…”

“…One such important connection to the interior point method literature was presented in the works of Kannan and Narayanan [6] and Narayanan [9] who proposed the Dikin walk in a polytope. Roughly, the uniform version of the Dikin walk, considered by [6], when at a point x ∈ K, computes the Dikin ellipsoid at x, and moves to a random point in it after a suitable Metropolis filter.…”

“…Several virtues of the Dikin ellipsoid (see [10,11,7]) were used by [6,9] to prove that the mixing time of the Dikin walk is O(mn) starting from a warm start, when K is described by m inequality constraints. Recall that a distribution over K is said to be a warm start if its density is bounded from above by a constant relative to the uniform distribution on K. Roughly, the proof (for either walk) consists of two parts: (1) an isoperimetric inequality, proved by Lovász [8], for convex bodies in terms of a distance introduced by Hilbert, and (2) a bound on the changes in the sampling distributions of the Dikin walk in terms of the Hilbert distance.…”

Sampling from Log-Concave Distributions with Infinity-Distance Guarantees

“…One line of work includes bounds for sampling from a log-concave distribution on a compactly supported convex body within TV distance O(δ), including results with running time that is polylogarithmic in 1 δ [1,30,29,34] (as well as other results which give a running time bound that is polynomial in 1 δ [17,16,4,3]). In addition to assuming access to a value oracle for f , some Markov chains just need access to a membership oracle for K [1,30,29], while others assume that K is a given polytope: {θ ∈ R d : Aθ ≤ b} [23,33,35,34,26]. They often also assume that K is contained in a ball of radius R and contains a ball of smaller radius r. Many of these results assume that the target log-concave distribution satisfies a "well-rounded" condition which says that the variance of the target distribution is Θ(d) [30,29], or that it is in isotropic position (that is, all the eigenvalues of its covariance matrix are Θ(1)) [25]; when applied to log-concave distributions that are not well-rounded or isotropic, these results require a "rounding" pre-processing procedure to find a linear transformation which makes the target distribution well-rounded or isotropic.…”

“…Finally, it is often assumed that the function f is such that f is L-Lipschitz or β-smooth [34], including works handling the widely-studied special case when f is uniform on K where L = β = 0 (see e.g. [28,23,33,35,26,6,21]).…”