# Quantile Regression for Large-scale Applications

;
JMLR W&CP 28
(3)
:
881–887, 2013

## Abstract

Quantile regression is a method to estimate the quantiles of the conditional distribution of a response variable, and as such it permits a much more accurate portrayal of the relationship between the response variable and observed covariates than methods such as Least-squares or Least Absolute Deviations regression. It can be expressed as a linear program, and interior-point methods can be used to find a solution for moderately large problems. Dealing with very large problems,

*e.g.*, involving data up to and beyond the terabyte regime, remains a challenge. Here, we present a randomized algorithm that runs in time that is nearly linear in the size of the input and that, with constant probability, computes a \((1+\epsilon)\) approximate solution to an arbitrary quantile regression problem. Our algorithm computes a low-distortion subspace-preserving embedding with respect to the loss function of quantile regression. Our empirical evaluation illustrates that our algorithm is competitive with the best previous work on small to medium-sized problems, and that it can be implemented in MapReduce-like environments and applied to terabyte-sized problems.