We analyze a stochastic approximation algorithm for decision-dependent problems, wherein the data distribution used by the algorithm evolves along the iterate sequence. The primary examples of such problems appear in performative prediction and its multiplayer extensions. We show that under mild assumptions, the deviation between the average iterate of the algorithm and the solution is asymptotically normal, with a covariance that clearly decouples the effects of the gradient noise and the distributional shift. Moreover, building on the work of H\'ajek and Le Cam, we show that the asymptotic performance of the algorithm with averaging is locally minimax optimal.

翻译：我们分析了一种用于决策依赖问题的随机逼近算法，其中算法使用的数据分布在迭代序列过程中演化。此类问题的主要实例出现在行为预测及其多智能体扩展中。我们证明，在温和假设下，算法平均迭代值与解之间的偏差是渐近正态的，其协方差清晰地将梯度噪声和分布偏移的影响解耦。此外，基于Hájek和Le Cam的研究，我们证明了该带平均的算法的渐近性能在局部上具有极小极大最优性。