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Given a matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ and a vector $b\in\mathbb{R}^{d}$, we show how to compute an $\epsilon$-approximate solutionto the regression problem $ \min_{x\in\mathbb{R}^{d}}\frac{1}{2} \|\mathbf{A} x- b\|_{2}^{2} $ in time $ \tilde{O} ((n+\sqrt{d\cdot\kappa_{\text{sum}}})\cdots\cdot\log\epsilon^{-1}) $ where$\kappa_{\text{sum}}=\mathrm{tr}\left(\mathbf{A}^{\top}\mathbf{A}\right)/\lambda_{\min}(\mathbf{A}^{T}\mathbf{A})$and $s$ is the maximum number of non-zero entries in a row of $\mathbf{A}$. Ouralgorithm improves upon the previous best running time of $ \tilde{O}((n+\sqrt{n \cdot\kappa_{\text{sum}}})\cdot s\cdot\log\epsilon^{-1})$.  We achieve our result through a careful combination of leverage scoresampling techniques, proximal point methods, and accelerated coordinatedescent. Our method not only matches the performance of previous methods, butfurther improves whenever leverage scores of rows are small (up topolylogarithmic factors). We also provide a non-linear generalization of theseresults that improves the running time for solving a broader class of ERMproblems.

Leverage Score Sampling for Faster Accelerated Regression and ERM