An Improved Cutting Plane Method for Convex Optimization, Convex-Concave Games and its Applications

Given a separation oracle for a convex set $K \subset \mathbb{R}^n$ that is contained in a box of radius $R$, the goal is to either compute a point in $K$ or prove that $K$ does not contain a ball of radius $ε$. We propose a new cutting plane algorithm that uses an optimal $O(n \log (κ))$ evaluations of the oracle and an additional $O(n^2)$ time per evaluation, where $κ= nR/ε$. $\bullet$ This improves upon Vaidya's $O( \text{SO} \cdot n \log (κ) + n^{ω+1} \log (κ))$ time algorithm [Vaidya, FOCS 1989a] in terms of polynomial dependence on $n$, where $ω< 2.373$ is the exponent of matrix multiplication and $\text{SO}$ is the time for oracle evaluation. $\bullet$ This improves upon Lee-Sidford-Wong's $O( \text{SO} \cdot n \log (κ) + n^3 \log^{O(1)} (κ))$ time algorithm [Lee, Sidford and Wong, FOCS 2015] in terms of dependence on $κ$. For many important applications in economics, $κ= Ω(\exp(n))$ and this leads to a significant difference between $\log(κ)$ and $\mathrm{poly}(\log (κ))$. We also provide evidence that the $n^2$ time per evaluation cannot be improved and thus our running time is optimal. A bottleneck of previous cutting plane methods is to compute leverage scores, a measure of the relative importance of past constraints. Our result is achieved by a novel multi-layered data structure for leverage score maintenance, which is a sophisticated combination of diverse techniques such as random projection, batched low-rank update, inverse maintenance, polynomial interpolation, and fast rectangular matrix multiplication. Interestingly, our method requires a combination of different fast rectangular matrix multiplication algorithms.