2013/05/01 by Yang, Jiyan, Meng, Xiangrui, Mahoney, Michael W.
#Data Structures and Algorithms (cs.DS) #Distributed #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Parallel #and Cluster Computing (cs.DC)
paper · doi:10.48550/arxiv.1305.0087
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, with appropriate preprocessing, interior-point methods can be used to find a solution for moderately large problems. Dealing with very large problems, \emph(e.g.), involving data up to and beyond the terabyte regime, remains a challenge. Here, we present a randomized algorithm that runs in nearly linear time in the size of the input and that, with constant probability, computes a (1+ε) approximate solution to an arbitrary quantile regression problem. As a key step, 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 in addition it can be implemented in MapReduce-like environments and applied to terabyte-sized problems.