Gaussian Cooling and $O^*(n^3)$ Algorithms for Volume and Gaussian Volume

Abstract

We present an $O^*(n^3)$ randomized algorithm for estimating the volume of a well-rounded convex body given by a membership oracle, improving on the previous best complexity of $O^*(n^4)$. The new algorithmic ingredient is an accelerated cooling schedule where the rate of cooling increases with the temperature. Previously, the known approach for potentially achieving this asymptotic complexity relied on a positive resolution of the KLS hyperplane conjecture, a central open problem in convex geometry. We also obtain an $O^*(n^3)$ randomized algorithm for integrating a standard Gaussian distribution over an arbitrary convex set containing the unit ball. Both the volume and Gaussian volume algorithms use an improved algorithm for sampling a Gaussian distribution restricted to a convex body. In this latter setting, as we show, the KLS conjecture holds and for a spherical Gaussian distribution with variance $\sigma^2$, the sampling complexity is $O^*(\max\{n^3, \sigma^2n^2\})$ for the first sample and $O^*(\max\{n^2, \sigma^2n^2\})$ for every subsequent sample.