Wasserstein Robust Performative Prediction via Lagrangian Relaxation

In machine learning, predictive models are trained on historical data. Their deployment may incentivize agents to strategically adapt their behavior, thereby inducing a model-dependent distribution shift. This phenomenon is known as performativity. This paper develops a Wasserstein distributionally robust framework for performative prediction, where the predictive model only has access to limited data. Using these data, we construct an ambiguity set centered on the empirical distribution, and optimize the predictive model against the worst-case distribution. Furthermore, we reformulate the objective as a tractable min-max optimization problem via Lagrangian relaxation, and allow the penalty to depend on the prediction model. Based on this, we develop distributionally robust repeated risk minimization (DR-RRM) and repeated gradient descent (DR-RGD) algorithms to iteratively find a performative stable point amid distributional shifts and model retraining. We theoretically show that both algorithms converge to a stable point linearly under standard regularity conditions. When accounting for approximation errors in the optimization problems, both algorithms converge to a neighborhood of the stable point. Additionally, we establish theoretical bounds on the suboptimality gap between the stable point and the global performative optimum.
Finally, numerical simulations of a dynamic credit scoring problem demonstrate the efficacy of the method.