Stochastic Saddle Point Problems with Decision-Dependent Distributions

• URL: http://arxiv.org/abs/2201.02313v1
• Date: Fri, 7 Jan 2022 03:36:41 GMT
• Title: Stochastic Saddle Point Problems with Decision-Dependent Distributions
• Authors: Killian Wood and Emiliano Dall'Anese
• Abstract summary: This paper focuses on saddle point problems with decision-dependent in both the static and time-varying settings. We introduce the notion of equilibrium points -- which are saddle points for the stationary minimax problem. We show that primal-dual algorithms converge to saddle points in a similar fashion.
• Score: 0.6091702876917279
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper focuses on stochastic saddle point problems with decision-dependent distributions in both the static and time-varying settings. These are problems whose objective is the expected value of a stochastic payoff function, where random variables are drawn from a distribution induced by a distributional map. For general distributional maps, the problem of finding saddle points is in general computationally burdensome, even if the distribution is known. To enable a tractable solution approach, we introduce the notion of equilibrium points -- which are saddle points for the stationary stochastic minimax problem that they induce -- and provide conditions for their existence and uniqueness. We demonstrate that the distance between the two classes of solutions is bounded provided that the objective has a strongly-convex-strongly-concave payoff and Lipschitz continuous distributional map. We develop deterministic and stochastic primal-dual algorithms and demonstrate their convergence to the equilibrium point. In particular, 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; moreover, we show convergence to a neighborhood in expectation and almost surely. Finally, we investigate a condition on the distributional map -- which we call opposing mixture dominance -- that ensures the objective is strongly-convex-strongly-concave. Under this assumption, we show that primal-dual algorithms converge to the saddle points in a similar fashion.

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