  Constructing valid prediction intervals rather than point estimates is a
well-established approach for uncertainty quantification in the regression
setting. Models equipped with this capacity output an interval of values in
which the ground truth target will fall with some prespecified probability.
This is an essential requirement in many real-world applications where simple
point predictions' inability to convey the magnitude and frequency of errors
renders them insufficient for high-stakes decisions. Quantile regression is a
leading approach for obtaining such intervals via the empirical estimation of
quantiles in the (non-parametric) distribution of outputs. This method is
simple, computationally inexpensive, interpretable, assumption-free, and
effective. However, it does require that the specific quantiles being learned
are chosen a priori. This results in (a) intervals that are arbitrarily
symmetric around the median which is sub-optimal for realistic skewed
distributions, or (b) learning an excessive number of intervals. In this work,
we propose Relaxed Quantile Regression (RQR), a direct alternative to quantile
regression based interval construction that removes this arbitrary constraint
whilst maintaining its strengths. We demonstrate that this added flexibility
results in intervals with an improvement in desirable qualities (e.g. mean
width) whilst retaining the essential coverage guarantees of quantile
regression.
