Event Date
SPEAKER: Wenxin Zhou; Assistant Professor, Mathematics, UC San Diego
TITLE: “Smoothed Quantile Regression: Fast Computation, Bootstrap Inference and Nonconvex Regularization”
ABSTRACT: Quantile regression methods provide a useful, complementary approach to classical least squares estimation of statistical models. Two primary features of quantile regression are (i) robustness against outliers in the response and (ii) the ability to capture heterogeneity. The lack of fast algorithms, however, often limits the use of quantile regression methods in practice. The past few decades have witnessed the development of the simplex algorithm, interior point algorithm and its variants for solving empirical quantile loss minimization. In the presence of large-scale datasets with unprecedented size and/or high dimensionality, fitting a single quantile regression using the state-of-the-art software is computationally prohibitive, let alone solving a sequence of quantile regressions at many quantile indices or performing bootstrap inference.
In this work, we study the smoothed quantile regression (SQR) proposed by Fernandes, Guerre and Horta (2019). First, we provide nonasymptotic results on the smoothing bias and statistical properties of the SQR estimator, which shed light on the choice of bandwidth as a function of dimension and sample size. Computationally, we apply the gradient descent method with a Barzilai-Borwein update step, using Huber regression as a warm start. The proposed algorithm is implemented via RcppArmadillo in R. Numerical experiments imply that without losing statistical accuracy, SQR-computations can be made considerably faster than standard QR-computations through the R package “quantreg". Lastly, we extend SQR regression to high dimensions, using both $\ell_1$- and nonconvex regularizations. For the latter, we apply the multi-stage convex relaxation method, an iterative scheme that constructs a sequence of solutions. Theoretically we show that optimal statistical properties can be achieved after a few iterations.
DATE: Thursday, October 17th, 4:10pm
LOCATION: MSB 1147, Colloquium Room
REFRESHMENTS: 3:30pm MSB 4110 (4th floor lounge)