Iterative Refinement for ℓ_p-norm Regression
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We give improved algorithms for the ℓ_p-regression problem, _xx_p such that A x=b, for all p ∈ (1,2) ∪ (2,∞). Our algorithms obtain a high accuracy solution in Õ_p(m^|p-2|/2p + |p-2|) <Õ_p(m^1/3) iterations, where each iteration requires solving an m × 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 ℓ_p-regression to 1 / poly(n) accuracy that run in time Õ_p(m^{ω, 7/3}), where ω is the matrix multiplication constant. For the current best value of ω > 2.37, we can thus solve ℓ_p regression as fast as ℓ_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 ℓ_p-norm flow / voltage solutions to 1 / poly(n) accuracy on an undirected graph with m edges in Õ_p(m^1 + |p-2|/2p + |p-2|) <Õ_p(m^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 ℓ_p-norms, using the smoothed ℓ_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 ℓ_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.
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