Online Kernel-Based Quantile Regression Using Huberized Pinball Loss

We present an efficient online kernel-based quantile regression scheme based on the Moreau envelope of the pinball loss, which we call the Huberized pinball loss. The use of the Moreau envelope is motivated by the popular Huber loss, which is the Moreau envelope of the least absolute deviation in robust estimation. We show that the smooth Huberized pinball loss exhibits more robust learning behaviours than the ordinary pinball loss in some scenarios, while the discrepancy of its minimizer from the true quantile is bounded by constants dependent on the Moreau-envelope parameter. Numerical examples show that the proposed scheme achieves better and more stable performances than a pinball-loss-based online method.