Non-Stationary Functional Bilevel Optimization

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.

Online Nonconvex Bilevel Optimization with Bregman Divergences

Bilevel optimization methods are increasingly relevant within machine learning, especially for tasks such as hyperparameter optimization and meta-learning. Compared to the offline setting, online bilevel optimization (OBO) offers a more dynamic framework by accommodating time-varying functions and sequentially arriving data. This study addresses the online nonconvex-strongly convex bilevel optimization problem. In deterministic settings, we introduce a novel online Bregman bilevel optimizer (OBBO) that utilizes adaptive Bregman divergences. We demonstrate that OBBO enhances the known sublinear rates for bilevel local regret through a novel hypergradient error decomposition that adapts to the underlying geometry of the problem. In stochastic contexts, we introduce the first stochastic online bilevel optimizer (SOBBO), which employs a window averaging method for updating outer-level variables using a weighted average of recent stochastic approximations of hypergradients. This approach not only achieves sublinear rates of bilevel local regret but also serves as an effective variance reduction strategy, obviating the need for additional stochastic gradient samples at each timestep. Experiments on online hyperparameter optimization and online meta-learning highlight the superior performance, efficiency, and adaptability of our Bregman-based algorithms compared to established online and offline bilevel benchmarks.