  Supervised learning is an active research area, with numerous applications in
diverse fields such as data analytics, computer vision, speech and audio
processing, and image understanding. In most cases, the loss functions used in
machine learning assume symmetric noise models, and seek to estimate the
unknown function parameters. However, loss functions such as quantile and
quantile Huber generalize the symmetric $\ell_1$ and Huber losses to the
asymmetric setting, for a fixed quantile parameter. In this paper, we propose
to jointly infer the quantile parameter and the unknown function parameters,
for the asymmetric quantile Huber and quantile losses. We explore various
properties of the quantile Huber loss and implement a convexity certificate
that can be used to check convexity in the quantile parameter. When the loss if
convex with respect to the parameter of the function, we prove that it is
biconvex in both the function and the quantile parameters, and propose an
algorithm to jointly estimate these. Results with synthetic and real data
demonstrate that the proposed approach can automatically recover the quantile
parameter corresponding to the noise and also provide an improved recovery of
function parameters. To illustrate the potential of the framework, we extend
the gradient boosting machines with quantile losses to automatically estimate
the quantile parameter at each iteration.
