Iterative Refinement for $\ell_p$-norm Regression
Deeksha Adil, Rasmus Kyng, Richard Peng, Sushant Sachdeva
TL;DR
The paper develops an iterative refinement framework for $\ell_p$-norm regression with $p\in(1,2)\cup(2,\infty)$, achieving geometric convergence by solving a sequence of smoothed residual problems. It provides two main algorithmic tracks: a fast residual solver for $p>2$ (and a dual strategy for $1<p<2$) that yields improved iteration counts, and a unified inverse-maintenance approach that delivers near-matrix-maximum-efficiency runtimes. When combined with fast graph Laplacian solvers, the framework yields near-optimal $\ell_p$-norm flow and voltage computations on graphs in time close to $m^{4/3}$ for constant $p$, marking a substantial speedup over prior methods. The results generalize to affine-norm variants and graph learning tasks, offering a broad, practical toolkit for high-accuracy $p$-norm optimization on large, structured problems.
Abstract
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 $\tilde{O}_p(m^{\max\{ω, 7/3\}}),$ where $ω$ is the matrix multiplication constant. For the current best value of $ω> 2.37$, 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.