Nearly Linear Row Sampling Algorithm for Quantile Regression

We give a row sampling algorithm for the quantile loss function with sample complexity nearly linear in the dimensionality of the data, improving upon the previous best algorithm whose sampling complexity has at least cubic dependence on the dimensionality. Based upon our row sampling algorithm, we give the fastest known algorithm for quantile regression and a graph sparsification algorithm for balanced directed graphs. Our main technical contribution is to show that Lewis weights sampling, which has been used in row sampling algorithms for p norms, can also be applied in row sampling algorithms for a variety of loss functions. We complement our theoretical results by experiments to demonstrate the practicality of our approach. p norms [14]. They show that by sampling O(d maxp/2+1,p+1 /ε 2 ) rows of A according to the p leverage scores, the resulting matrix A satisfies (1 -ε) Ax p ≤ A x p ≤ (1 + ε) Ax p 1 Throughout the paper, we use O(f ) to denote O(f polylog f ).