Computing Lewis weights to high precision using local relative smoothness

We provide algorithms that compute ε-estimates of the ℓp-Lewis weights of a matrix A ∈ ℝm × n for p ≥ 4 using O(p2 log(m/ε)) rounds of leverage score computation, where ℓp-Lewis weights and leverage scores are both standard measures of row importance. This improves upon the state-of-the-art round complexity of O(p3 log(m/ε)) due to Fazel, Lee, Padmanabha, and Sidford (2022). We obtain our results by carefully applying a local variant of relatively smooth gradient descent to primal and dual forms of the ℓp-Lewis weight optimization problem and providing tools to convert between different notions of approximate ℓp-Lewis weights.