Performative prediction is a recently proposed framework where predictions guide decision-making and hence influence future data distributions. Such performative phenomena are ubiquitous in various areas, such as transportation, finance, public policy, and recommendation systems. To date, work on performative prediction has only focused on unconstrained scenarios, neglecting the fact that many real-world learning problems are subject to constraints. This paper bridges this gap by studying performative prediction under inequality constraints. Unlike most existing work that provides only performative stable points, we aim to find the optimal solutions. Anticipating performative gradients is a challenging task, due to the agnostic performative effect on data distributions. To address this issue, we first develop a robust primal-dual framework that requires only approximate gradients up to a certain accuracy, yet delivers the same order of performance as the stochastic primal-dual algorithm without performativity. Based on this framework, we then propose an adaptive primal-dual algorithm for location families. Our analysis demonstrates that the proposed adaptive primal-dual algorithm attains $\ca{O}(\sqrt{T})$ regret and constraint violations, using only $\sqrt{T} + 2T$ samples, where $T$ is the time horizon. To our best knowledge, this is the first study and analysis on the optimality of the performative prediction problem under inequality constraints. Finally, we validate the effectiveness of our algorithm and theoretical results through numerical simulations.

翻译：表演性预测是一个近期提出的框架，其中预测指导决策，进而影响未来数据分布。此类表演性现象在交通、金融、公共政策和推荐系统等多个领域普遍存在。迄今为止，关于表演性预测的研究仅聚焦于无约束场景，忽视了现实世界中的许多学习问题均受限于约束条件。本文通过研究不等式约束下的表演性预测填补了这一空白。与大多数仅提供表演性稳定点的现有工作不同，我们旨在寻找最优解。由于数据分布的表演性效应不可知，预测表演性梯度是一项具有挑战性的任务。为解决这一问题，我们首先开发了一个鲁棒的原-对偶框架，该框架仅需达到特定精度的近似梯度，却能提供与无表演性随机原-对偶算法同量级的性能。基于该框架，我们随后为位置族提出了一种自适应原-对偶算法。分析表明，所提出的自适应原-对偶算法在时间范围T内，仅使用√T + 2T个样本即可实现O(√T)的遗憾值和约束违反。据我们所知，这是首次对不等式约束下表演性预测问题的最优性进行研究和分析。最后，我们通过数值模拟验证了算法与理论结果的有效性。