  When predictions are performative, the choice of which predictor to deploy
influences the distribution of future observations. The overarching goal in
learning under performativity is to find a predictor that has low
\emph{performative risk}, that is, good performance on its induced
distribution. One family of solutions for optimizing the performative risk,
including bandits and other derivative-free methods, is agnostic to any
structure in the performative feedback, leading to exceedingly slow convergence
rates. A complementary family of solutions makes use of explicit \emph{models}
for the feedback, such as best-response models in strategic classification,
enabling faster rates. However, these rates critically rely on the feedback
model being correct. In this work we study a general protocol for making use of
possibly misspecified models in performative prediction, called \emph{plug-in
performative optimization}. We show this solution can be far superior to
model-agnostic strategies, as long as the misspecification is not too extreme.
Our results support the hypothesis that models, even if misspecified, can
indeed help with learning in performative settings.
