  In this paper, we propose a general distributionally robust framework for
performative optimization, where the selected decision can influence the
probabilistic distribution of uncertain parameters. Our framework facilitates
safe decision-making in scenarios with incomplete information about the
underlying decision-dependent distributions, relying instead on accessible
reference distributions. To tackle the challenge of decision-dependent
uncertainty, we introduce an algorithm named repeated robust risk minimization.
This algorithm decouples the decision variables associated with the ambiguity
set from the expected loss, optimizing the latter at each iteration while
keeping the former fixed to the previous decision. By leveraging the strong
connection between distributionally robust optimization and regularization, we
establish a linear convergence rate to a performatively stable point and
provide a suboptimality performance guarantee for the proposed algorithm.
Finally, we examine the performance of our proposed model through an
experimental study in strategic classification.
