Faster polytope rounding, sampling, and volume computation via a sublinear "Ball Walk"

We study the problem of "isotropically rounding" a polytope $K\subset\mathbb{R}^n$, that is, computing a linear transformation which makes the uniform distribution on the polytope have roughly identity covariance matrix. We assume $K$ is defined by $m$ linear inequalities, with guarantee that $rB\subset K\subset RB$, where $B$ is the unit ball. We introduce a new variant of the ball walk Markov chain and show that, roughly, the expected number of arithmetic operations per-step of this Markov chain is $O(m)$ that is sublinear in the input size $mn$--the per-step time of all prior Markov chains. Subsequently, we give a rounding algorithm that succeeds with probability $1-\varepsilon$ in $\tilde{O}(mn^{4.5}\mbox{polylog}(\frac{1}{\varepsilon},\frac{R}{r}))$ arithmetic operations. This gives a factor of $\sqrt{n}$ improvement on the previous bound of $\tilde{O}(mn^5\mbox{polylog}(\frac{1}{\varepsilon},\frac{R}{r}))$ for rounding, which uses the hit-and-run algorithm. Since the rounding preprocessing step is in many cases the bottleneck in improving sampling or volume computation, our results imply these tasks can also be achieved in roughly $\tilde{O}(mn^{4.5}\mbox{polylog}(\frac{1}{\varepsilon},\frac{R}{r})+mn^4δ^{-2})$ operations for computing the volume of $K$ up to a factor $1+δ$ and $\tilde{O}(mn^{4.5}\mbox{polylog}(\frac{1}{\varepsilon},\frac{R}{r})))$ for uniformly sampling on $K$ with TV error $\varepsilon$. This improves on the previous bounds of $\tilde{O}(mn^5\mbox{polylog}(\frac{1}{\varepsilon},\frac{R}{r})+mn^4δ^{-2})$ for volume computation when roughly $m\geq n^{2.5}$, and $\tilde{O}(mn^5\mbox{polylog}(\frac{1}{\varepsilon},\frac{R}{r}))$ for sampling when roughly $m\geq n^{1.5}$. We achieve this improvement by a novel method of computing polytope membership, where one avoids checking inequalities estimated to have a very low probability of being violated.