Performative Prediction in the Wild: Adapting to Arbitrary Data Distribution Maps

Abstract Performative prediction refers to scenarios where model predictions influence the underlying data distribution they aim to predict. A desirable property in this context is performative stability , where model predictions are already optimal for the distribution they induce, indicating converged model parameters and no need for further retraining. Achieving performative stability requires characterizing the data distribution map $$\mathcal {D}(\theta )$$ , i.e., the relationship between predictions and the resulting distribution shifts. Current studies typically quantify distribution differences using metrics like $$\mathcal {W}_1$$ distance or $$\chi ^2$$ divergence, which may not provide isometric embeddings or maintain metric equivalence in practical scenarios, limiting their applicability across various data distribution maps. Moreover, the crucial smoothness parameter $$\beta $$ in existing work is often unobtainable in performative scenarios, constraining the real-world utility of current theoretical results and methods. To address these challenges, we develop an algorithm that learns a performatively stable model for arbitrary data distribution maps without requiring the joint smoothness parameter $$\beta $$ . Specifically, we introduce a new $$\hat{\varepsilon }$$ -sensitivity measure for $$\mathcal {D}(\theta )$$ , quantified by the gradient of the loss function, which naturally and directly characterizes how distribution shifts affect the optimization of the objective function. Based on this sensitivity, we formulate a $$\gamma $$ -strongly convex loss function and optimize the deployed model accordingly, where $$\gamma $$ is derived from the defined $$\hat{\varepsilon }$$ , eliminating the need for the $$\beta $$ -joint smoothness assumption. Our theoretical results guarantee the convergence of the deployed model to performative stability. Extensive experiments on synthetic and real-world datasets with diverse data distribution maps demonstrate the superiority of our method over state-of-the-art techniques in two key aspects: prediction accuracy and performative stability.