  In performative prediction, a predictive model impacts the distribution that
generates future data, a phenomenon that is being ignored in classical
supervised learning. In this closed-loop setting, the natural measure of
performance named performative risk ($\mathrm{PR}$), captures the expected loss
incurred by a predictive model \emph{after} deployment. The core difficulty of
using the performative risk as an optimization objective is that the data
distribution itself depends on the model parameters. This dependence is
governed by the environment and not under the control of the learner. As a
consequence, even the choice of a convex loss function can result in a highly
non-convex $\mathrm{PR}$ minimization problem. Prior work has identified a pair
of general conditions on the loss and the mapping from model parameters to
distributions that implies the convexity of the performative risk. In this
paper, we relax these assumptions and focus on obtaining weaker notions of
convexity, without sacrificing the amenability of the $\mathrm{PR}$
minimization problem for iterative optimization methods.
