Stochastic Saddle Point Problems with Decision-Dependent Distributions
本文研究目标函数分布随决策变量变化的随机鞍点问题，提出均衡点概念并给出存在唯一性条件，开发了确定性和随机原始对偶算法，在无分布映射先验知识时提出对立混合主导条件确保目标强凸强凹，并证明无导数算法可近似鞍点。
.This paper focuses on stochastic saddle point problems with decision-dependent distributions. These are problems whose objective is the expected value of a stochastic payoff function and whose data distribution drifts in response to decision variables—a phenomenon represented by a distributional map. A common approach to accommodating distributional shift is to retrain optimal decisions once a new distribution is revealed, or repeated retraining. We introduce the notion of equilibrium points, which are the fixed points of this repeated retraining procedure, and provide sufficient conditions for their existence and uniqueness. To find equilibrium points, we develop deterministic and stochastic primal-dual algorithms and demonstrate their convergence with constant step size in the former and polynomial decay step-size schedule in the latter. By modeling errors emerging from a stochastic gradient estimator as sub-Weibull random variables, we provide error bounds in expectation and in high probability that hold for each iteration. Without additional knowledge of the distributional map, computing saddle points is intractable. Thus we propose a condition on the distributional map—which we call opposing mixture dominance—that ensures that the objective is strongly-convex–strongly-concave. Finally, we demonstrate that derivative-free algorithms with a single function evaluation are capable of approximating saddle points.Keywordssaddle point problemsminimax problemsstochastic gradientdecision-dependent distributionsMSC codes90C2590C1590C47