Optimizing the Performative Risk under Weak Convexity Assumptions

In performative prediction, a predictive model impacts the distribution that generates future data, a phenomenon that is being ignored in classical supervised learning. In this closed-loop setting, the natural measure of performance, denoted the performative risk, captures the expected loss incurred by a predictive model after deployment. The core difficulty of minimizing the performative risk is that the data distribution itself depends on the model parameters. This dependence is governed by the environment and not under the control of the learner. As a consequence, even the choice of a convex loss function can result in a highly non-convex performative risk minimization problem. Prior work has identified a pair of general conditions on the loss and the mapping from model parameters to distributions that implies convexity of the performative risk. In this paper, we relax these assumptions and focus on obtaining weaker notions of convexity, without sacrificing the amenability of the performative risk minimization problem for iterative optimization methods.