Adaptive Stochastic Gradient Descent Ascent Algorithm for Nonconvex Minimax Problems with Decision-Dependent Distributions

In this paper, we study stochastic minimax problems with decision-dependent distributions (SMDD), where the probability distribution of stochastic variable depends on decision variable. For SMDD with nonconvex-(strongly) concave objective function, we propose an adaptive stochastic gradient descent ascent algorithm (ASGDA) to find the stationary points of SMDD, which learns the unknown distribution map dynamically and optimizes the minimax problem simultaneously. When the distribution map follows a location-scale model, we show that ASGDA finds an $\epsilon$-stationary point within $\mathcal{O}\left(\epsilon^{-\left(4+\delta\right)} \right)$ for $\forall\delta>0$, and $\mathcal{O}(\epsilon^{-8})$ stochastic gradient evaluations in nonconvex-strongly concave and nonconvex-concave settings respectively. When the objective function of SMDD is nonconvex in $x$ and satisfies Polyak-{\L}ojasiewicz (P{\L}) inequality in $y$, we propose an alternating adaptive stochastic gradient descent ascent algorithm (AASGDA) and show that AASGDA finds an $\epsilon$-stationary point within $\mathcal{O}(\kappa_y^4\epsilon^{-4})$ stochastic gradient evaluations, where $\kappa_y$ denotes the condition number. We verify the effectiveness of the proposed algorithms through numerical experiments on both synthetic and real-world data.