Practical Volume Estimation of Zonotopes by a New Annealing Schedule for Cooling Convex Bodies

Abstract

We study the problem of estimating the volume of convex polytopes, focusing on zonotopes. Although a lot of effort is devoted to practical algorithms for polytopes given as an intersection of halfspaces, there is no such method for zonotopes. Our algorithm is based on Multiphase Monte Carlo (MMC) methods, and our main contributions include: (i) a new uniform sampler employing Billiard Walk for the first time in volume computation, (ii) a new simulated annealing generalizing existing MMC by making use of adaptive convex bodies which fit to the input, thus drastically reducing the number of phases. Extensive experiments on zonotopes show our algorithm requires sub-linear number of oracle calls in the dimension, while the best theoretical bound is cubic. Moreover, our algorithm can be easily generalized to any convex body. We offer an open-source, optimized C++ implementation, and analyze its performance. Our code tackles problems intractable so far, offering the first efficient algorithm for zonotopes which scales to high dimensions (e.g. one hundred dimensions in less than 1 h).