Fast learning rates for sparse quantile regression problem

Learning with coefficient-based regularization has attracted a considerable amount of attention recently in both machine learning and statistics. This paper presents a kernelized version of a quantile estimator integrated with coefficient-based regularization, which can be solved efficiently by a simple linear programming. Fast convergence rates are obtained under mild condition on the underlying distribution. Besides, this algorithm can be adapted easily to large scale problems and sparse solution is often achieved as that of Lasso. In our work we make the following main contributions: girst, improved learning rates are obtained by employing so called variance bounds, which is optimal in the literatures of learning theory; second, we establish stronger convergence rates by employing self-calibration inequalities; third, our learning rates can also be derived by a simple data-dependent parameter selection method; finally, the performance of the classical and our new algorithms are compared respectively in a simulation study and an actual problem.