Trust Region Algorithm for Stochastic Minimax Problems with Decision-Dependent Distributions

Abstract: Stochastic minimax optimization has drawn much attention over the past decade due to its broad applications in machine learning, signal processing and game theory. In some applications, the probability distribution of uncertainty depends on decision variables, as the environment may respond to decisions. In this paper, we propose a trust region algorithm for finding the stationary points of stochastic minimax problems with decision-dependent distributions. At each iteration, the algorithm locally learns the dependence of the random variable on the decision variable from data samples via linear regression, and updates the decision variable by solving trust-region subproblem with the learned distribution. When the objective function is nonconvex--strongly concave and the distribution map follows a regression model, we prove the almost sure convergence of the iterates to a stationary point of the primal function. The effectiveness of the proposed algorithm is further demonstrated through numerical experiments on both synthetic and real-world data sets.