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：我们分析了针对决策依赖问题的随机逼近算法，其中算法所使用的数据分布沿迭代序列演化。此类问题的主要范例出现在执行预测（performative prediction）及其多人扩展中。我们证明，在温和假设下，算法平均迭代量与解之间的偏差具有渐近正态性，其协方差清晰地解耦了梯度噪声与分布偏移的影响。此外，基于Hájek和Le Cam的研究，我们证明带平均处理的算法的渐近性能具有局部极小极大最优性。