  Constrained learning has become increasingly important, especially in the
realm of algorithmic fairness and machine learning. In these settings,
predictive models are developed specifically to satisfy pre-defined notions of
fairness. Here, we study the general problem of constrained statistical machine
learning through a statistical functional lens. We consider learning a
function-valued parameter of interest under the constraint that one or several
pre-specified real-valued functional parameters equal zero or are otherwise
bounded. We characterize the constrained functional parameter as the minimizer
of a penalized risk criterion using a Lagrange multiplier formulation. We show
that closed-form solutions for the optimal constrained parameter are often
available, providing insight into mechanisms that drive fairness in predictive
models. Our results also suggest natural estimators of the constrained
parameter that can be constructed by combining estimates of unconstrained
parameters of the data generating distribution. Thus, our estimation procedure
for constructing fair machine learning algorithms can be applied in conjunction
with any statistical learning approach and off-the-shelf software. We
demonstrate the generality of our method by explicitly considering a number of
examples of statistical fairness constraints and implementing the approach
using several popular learning approaches.
