  As data-driven methods are deployed in real-world settings, the processes
that generate the observed data will often react to the decisions of the
learner. For example, a data source may have some incentive for the algorithm
to provide a particular label (e.g. approve a bank loan), and manipulate their
features accordingly. Work in strategic classification and decision-dependent
distributions seeks to characterize the closed-loop behavior of deploying
learning algorithms by explicitly considering the effect of the classifier on
the underlying data distribution. More recently, works in performative
prediction seek to classify the closed-loop behavior by considering general
properties of the mapping from classifier to data distribution, rather than an
explicit form. Building on this notion, we analyze repeated risk minimization
as the perturbed trajectories of the gradient flows of performative risk
minimization. We consider the case where there may be multiple local minimizers
of performative risk, motivated by situations where the initial conditions may
have significant impact on the long-term behavior of the system. We provide
sufficient conditions to characterize the region of attraction for the various
equilibria in this settings. Additionally, we introduce the notion of
performative alignment, which provides a geometric condition on the convergence
of repeated risk minimization to performative risk minimizers.
