On the nonconvexity of some push-forward constraints and its
  consequences in machine learning
Lucas de Lara Mathis Deronzier Alberto González-Sanz Virgile Foy
The push-forward operation enables one to redistribute a probability measure
through a deterministic map. It plays a key role in statistics and
optimization: many learning problems (notably from optimal transport,
generative modeling, and algorithmic fairness) include constraints or penalties
framed as push-forward conditions on the model. However, the literature lacks
general theoretical insights on the (non)convexity of such constraints and its
consequences on the associated learning problems. This paper aims at filling
this gap. In a first part, we provide a range of sufficient and necessary
conditions for the (non)convexity of two sets of functions: the maps
transporting one probability measure to another; the maps inducing equal output
distributions across distinct probability measures. This highlights that for
most probability measures, these push-forward constraints are not convex. In a
second time, we show how this result implies critical limitations on the design
of convex optimization problems for learning generative models or group-fair
predictors. This work will hopefully help researchers and practitioners have a
better understanding of the critical impact of push-forward conditions onto
convexity.