Nearly Linear Row Sampling Algorithm for Quantile Regression
This work provides a faster algorithm for quantile regression and graph sparsification, addressing computational bottlenecks in machine learning and data analysis, though it is incremental as it extends existing Lewis weights sampling to new loss functions.
The paper tackles the problem of row sampling for quantile regression by introducing an algorithm with nearly linear sample complexity in data dimensionality, improving upon previous cubic dependencies, and demonstrates its practicality through experiments.
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 $\ell_p$ 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.