  This paper introduces a framework for estimating fair optimal predictions
using machine learning where the notion of fairness can be quantified using
path-specific causal effects. We use a recently developed approach based on
Lagrange multipliers for infinite-dimensional functional estimation to derive
closed-form solutions for constrained optimization based on mean squared error
and cross-entropy risk criteria. The theoretical forms of the solutions are
analyzed in detail and described as nuanced adjustments to the unconstrained
minimizer. This analysis highlights important trade-offs between risk
minimization and achieving fairnes. The theoretical solutions are also used as
the basis for construction of flexible semiparametric estimation strategies for
these nuisance components. We describe the robustness properties of our
estimators in terms of achieving the optimal constrained risk, as well as in
terms of controlling the value of the constraint. We study via simulation the
impact of using robust estimators of pathway-specific effects to validate our
theory. This work advances the discourse on algorithmic fairness by integrating
complex causal considerations into model training, thus providing strategies
for implementing fair models in real-world applications.
