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 nicely 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 is locally minimax optimal.

翻译：我们分析一个决定依赖性问题的随机近似算法,其中算法使用的数据分布沿周期序列演变。这些问题的主要例子出现在性能预测及其多玩家延伸中。我们表明,在轻度假设下,算法平均迭代与解决办法之间的偏差是无常的,其共变性可以很好地分离梯度噪音和分布式变化的影响。此外,在H\'ajek和Le Cam的工作基础上,我们证明算法的无药可治性表现是当地最理想的。