Iterative Refinement for $\ell_p$-norm Regression

We give improved algorithms for the $\ell{p}$-regression problem, $\min{x} |x|{p}$ such that $A x=b,$ for all $p \in (1,2) \cup (2,\infty).$ Our algorithms obtain a high accuracy solution in $\tilde{O}{p}(m^{{\frac{|p-2|}{2p} + |p-2|}}) \le \tilde{O}{p}(m^{{\frac{1}{3}})$} iterations, where each iteration requires solving an $m \times m$ linear system, $m$ being the dimension of the ambient space. By maintaining an approximate inverse of the linear systems that we solve in each iteration, we give algorithms for solving $\ell{p}$-regression to $1 / \text{poly}(n)$ accuracy that run in time $\tilde{O}p(m^{{\max{\omega,} 7/3}}),$ where $\omega$ is the matrix multiplication constant. For the current best value of $\omega > 2.37$, we can thus solve $\ell{p}$ regression as fast as $\ell{2}$ regression, for all constant $p$ bounded away from $1.$ Our algorithms can be combined with fast graph Laplacian linear equation solvers to give minimum $\ell{p}$-norm flow / voltage solutions to $1 / \text{poly}(n)$ accuracy on an undirected graph with $m$ edges in $\tilde{O}{p}(m^{1 + \frac{|p-2|}{2p + |p-2|}}) \le \tilde{O}{p}(m^{{\frac{4}{3}})$} time. For sparse graphs and for matrices with similar dimensions, our iteration counts and running times improve on the $p$-norm regression algorithm by [Bubeck-Cohen-Lee-Li STOC`18] and general-purpose convex optimization algorithms. At the core of our algorithms is an iterative refinement scheme for $\ell{p}$-norms, using the smoothed $\ell{p}$-norms introduced in the work of Bubeck et al. Given an initial solution, we construct a problem that seeks to minimize a quadratically-smoothed $\ell_{p}$ norm over a subspace, such that a crude solution to this problem allows us to improve the initial solution by a constant factor, leading to algorithms with fast convergence.