Efficient Sampling from Feasible Sets of SDPs and Volume Approximation

TL;DR

This paper introduces new algorithms based on geometric random walks for efficiently sampling from spectrahedra, enabling faster volume approximation and expectation computation in high-dimensional semidefinite programming problems.

Contribution

The paper develops and implements novel random walk algorithms that outperform existing methods in sampling from spectrahedra, with applications to volume estimation and robust control.

Findings

Faster mixing times than traditional Hit-and-Run methods.

Efficient sampling in dimensions up to 200.

Open source C++ implementation demonstrating scalability.

Abstract

We present algorithmic, complexity, and implementation results on the problem of sampling points from a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is geometric random walks. We analyze the arithmetic and bit complexity of certain primitive geometric operations that are based on the algebraic properties of spectrahedra and the polynomial eigenvalue problem. This study leads to the implementation of a broad collection of random walks for sampling from spectrahedra that experimentally show faster mixing times than methods currently employed either in theoretical studies or in applications, including the popular family of Hit-and-Run walks. The different random walks offer a variety of advantages , thus allowing us to efficiently sample from general probability distributions, for example the family of log-concave distributions which arise in numerous applications. We focus on two major applications of independent interest: (i) approximate the volume of a spectrahedron, and (ii) compute the expectation of functions coming from robust optimal control. We exploit efficient linear algebra algorithms and implementations to address the aforemen-tioned computations in very high dimension. In particular, we provide a C++ open source implementation of our methods that scales efficiently, for the first time, up to dimension 200. We illustrate its efficiency on various data sets.