Leverage Score Sampling for Faster Accelerated Regression and ERM

Given a matrix $\mathbf{A}\in {\mathbb{R}}^{n\times d}$ and a vector $b \in$\mathbb{R}$^{d}$,weshowhowtocomputean$\epsilon$-approximate solution to the regression problem $ \min_{x\in$\mathbb{R}$^{d}}\frac{1}{2} |$\mathbf{A}$ x

• b|{2}^{2} $intime$ \tilde{O} ((n+\sqrt{d\cdot\kappa{$\text{sum}$}})\cdot s\cdot\log\epsilon^{-1}) $where$\kappa_{$\text{sum}$}=$\mathrm{t}\mathrm{r}$\left($\mathbf{A}$^{\top}$\mathbf{A}$\right)/\lambda_{\min}($\mathbf{A}$^{T}$\mathbf{A}$)$and$s$isthemaximumnumberofnon-zeroentriesinarowof$$\mathbf{A}$$. Our algorithm 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 score sampling techniques, proximal point methods, and accelerated coordinate descent. Our method not only matches the performance of previous methods, but further improves whenever leverage scores of rows are small (up to polylogarithmic factors). We also provide a non-linear generalization of these results that improves the running time for solving a broader class of ERM problems.