  Performative prediction is a framework for learning models that influence the
data they intend to predict. We focus on finding classifiers that are
performatively stable, i.e. optimal for the data distribution they induce.
Standard convergence results for finding a performatively stable classifier
with the method of repeated risk minimization assume that the data distribution
is Lipschitz continuous to the model's parameters. Under this assumption, the
loss must be strongly convex and smooth in these parameters; otherwise, the
method will diverge for some problems. In this work, we instead assume that the
data distribution is Lipschitz continuous with respect to the model's
predictions, a more natural assumption for performative systems. As a result,
we are able to significantly relax the assumptions on the loss function. In
particular, we do not need to assume convexity with respect to the model's
parameters. As an illustration, we introduce a resampling procedure that models
realistic distribution shifts and show that it satisfies our assumptions. We
support our theory by showing that one can learn performatively stable
classifiers with neural networks making predictions about real data that shift
according to our proposed procedure.
