Computing Lewis weights to high precision using local relative smoothness
This work improves the computational efficiency of computing ℓ_p-Lewis weights, which are important for row sampling in linear algebra and machine learning, by reducing the number of leverage score computations needed.
The authors present algorithms that compute ε-estimates of ℓ_p-Lewis weights for p ≥ 4 with O(p² log(m/ε)) rounds of leverage score computation, improving upon the previous state-of-the-art round complexity of O(p³ log(m/ε)).
We provide algorithms that compute $ε$-estimates of the $\ell_p$-Lewis weights of a matrix $A \in \mathbb{R}^{m \times n}$ for $p \geq 4$ using $O(p^2 \log(m/ε))$ rounds of leverage score computation, where $\ell_p$-Lewis weights and leverage scores are both standard measures of row importance. This improves upon the state-of-the-art round complexity of $O(p^3 \log(m/ε))$ 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 $\ell_p$-Lewis weight optimization problem and providing tools to convert between different notions of approximate $\ell_p$-Lewis weights.