Functional bilevel optimization (FBO) provides a powerful framework for hierarchical learning in function spaces, yet current methods are limited to static offline settings and perform suboptimally in online, non-stationary scenarios. We propose SmoothFBO, the first algorithm for non-stationary FBO with both theoretical guarantees and practical scalability. SmoothFBO introduces a time-smoothed stochastic hypergradient estimator that reduces variance through a window parameter, enabling stable outer-loop updates with sublinear regret. Importantly, the classical parametric bilevel case is a special reduction of our framework, making SmoothFBO a natural extension to online, non-stationary settings. Empirically, SmoothFBO consistently outperforms existing FBO methods in non-stationary hyperparameter optimization and model-based reinforcement learning, demonstrating its practical effectiveness. Together, these results establish SmoothFBO as a general, theoretically grounded, and practically viable foundation for bilevel optimization in online, non-stationary scenarios.

翻译：函数双层优化为函数空间中的分层学习提供了强大框架，但现有方法仅限于静态离线设置，在在线非平稳场景中表现欠佳。我们提出了SmoothFBO，这是首个兼具理论保证与实践可扩展性的非平稳函数双层优化算法。SmoothFBO引入了一种时间平滑的随机超梯度估计器，通过窗口参数降低方差，从而实现具有次线性遗憾的稳定外层更新。重要的是，经典参数化双层情形是我们框架的特殊约化形式，使得SmoothFBO自然扩展至在线非平稳场景。实证表明，在非平稳超参数优化和基于模型的强化学习中，SmoothFBO持续优于现有函数双层优化方法，证明了其实际有效性。这些结果共同确立了SmoothFBO作为在线非平稳场景中双层优化的通用、理论坚实且实践可行的基础。