Coordinate Methods for Accelerating $\ell_\infty$ Regression and Faster Approximate Maximum Flow

We provide faster algorithms for approximately solving $\ell_{\infty}$ regression, a fundamental problem prevalent in both combinatorial and continuous optimization. In particular, we provide accelerated coordinate descent methods capable of provably exploiting dynamic measures of coordinate smoothness, and apply them to $\ell_\infty$ regression over a box to give algorithms which converge in $k$ iterations at a $O(1/k)$ rate. Our algorithms can be viewed as an alternative approach to the recent breakthrough result of Sherman [She17] which achieves a similar runtime improvement over classic algorithmic approaches, i.e. smoothing and gradient descent, which either converge at a $O(1/\sqrt{k})$ rate or have running times with a worse dependence on problem parameters. Our runtimes match those of [She17] across a broad range of parameters and achieve improvement in certain structured cases. We demonstrate the efficacy of our result by providing faster algorithms for the well-studied maximum flow problem. Directly leveraging our accelerated $\ell_\infty$ regression algorithms imply a $\tilde{O}\left(m + \sqrt{mn}/\epsilon\right)$ runtime to compute an $\epsilon$-approximate maximum flow for an undirected graph with $m$ edges and $n$ vertices, generically improving upon the previous best known runtime of $\tilde{O}\left(m/\epsilon\right)$ in [She17] whenever the graph is slightly dense. We further design an algorithm adapted to the structure of the regression problem induced by maximum flow obtaining a runtime of $\tilde{O}\left(m + \max(n, \sqrt{ns})/\epsilon\right)$, where $s$ is the squared $\ell_2$ norm of the congestion of any optimal flow. Moreover, we show how to leverage this result to achieve improved exact algorithms for maximum flow on a variety of unit capacity graphs. We hope that our work serves as an important step towards achieving even faster maximum flow algorithms.