In machine learning, training data often capture the behaviour of multiple
subgroups of some underlying human population. When the nature of training data
for subgroups are not controlled carefully, under-representation bias arises.
To counter this effect we introduce two natural notions of subgroup fairness
and instantaneous fairness to address such under-representation bias in
time-series forecasting problems. Here we show globally convergent methods for
the fairness-constrained learning problems using hierarchies of
convexifications of non-commutative polynomial optimisation problems. Our
empirical results on a biased data set motivated by insurance applications and
the well-known COMPAS data set demonstrate the efficacy of our methods. We also
show that by exploiting sparsity in the convexifications, we can reduce the run
time of our methods considerably.Comment: Journal version of Zhou et al. [arXiv:2006.07315, AAAI 2021