Online Saddle Point Tracking with Decision-Dependent Data

Abstract: In this work, we consider a time-varying stochastic saddle point problem in which the objective is revealed sequentially, and the data distribution depends on the decision variables. Problems of this type express the distributional dependence via a distributional map, and are known to have two distinct types of solutions--saddle points and equilibrium points. We demonstrate that, under suitable conditions, online primal-dual type algorithms are capable of tracking equilibrium points. In contrast, since computing closed-form gradient of the objective requires knowledge of the distributional map, we offer an online stochastic primal-dual algorithm for tracking equilibrium trajectories. We provide bounds in expectation and in high probability, with the latter leveraging a sub-Weibull model for the gradient error. We illustrate our results on an electric vehicle charging problem where responsiveness to prices follows a location-scale family based distributional map.