  This paper considers doing quantile regression on censored data using neural
networks (NNs). This adds to the survival analysis toolkit by allowing direct
prediction of the target variable, along with a distribution-free
characterisation of uncertainty, using a flexible function approximator. We
begin by showing how an algorithm popular in linear models can be applied to
NNs. However, the resulting procedure is inefficient, requiring sequential
optimisation of an individual NN at each desired quantile. Our major
contribution is a novel algorithm that simultaneously optimises a grid of
quantiles output by a single NN. To offer theoretical insight into our
algorithm, we show firstly that it can be interpreted as a form of
expectation-maximisation, and secondly that it exhibits a desirable
`self-correcting' property. Experimentally, the algorithm produces quantiles
that are better calibrated than existing methods on 10 out of 12 real datasets.
