Bilevel Optimization with Coupled Decision-Dependent Distributions

Bilevel optimization has gained significant popularity in recent years due to its ability to formulate various machine learning problems. For instance, in meta-learning, the upper-level (UL) problem offers a good initialization for the lower-level (LL) model to facilitate adaptation. However, the decision variables can impact data features and outcomes, leading to the phenomenon known as performativity. In this work, we investigate the inclusion of decision-dependent distributions in bilevel optimization. Specifically, we consider the scenarios where the UL data distribution depends on the LL optimization variable, and the LL data distribution also depends on the UL decision variable. We first establish sufficient conditions for the existence of performatively stable (PS) solutions in this class of bilevel problems. Also, we propose efficient stochastic algorithms to find the PS point with theoretical convergence rate analysis and discuss the theoretical optimality of the obtained solution. Our theoretical analysis is corroborated through a series of numerical experiments, wherein we evaluate the performance of the bilevel performative prediction algorithms alongside non-performative counterparts in the context of meta strategic learning problems.