Online Bilevel Optimization: Regret Analysis of Online Alternating Gradient Methods

This paper introduces $\mathit{\text{online}}\mathit{\text{bilevel}}\mathit{\text{optimization}}$ in which a sequence of time-varying bilevel problems is revealed one after the other. We extend the known regret bounds for online single-level algorithms to the bilevel setting. Specifically, we provide new notions of $\mathit{\text{bilevel}}\mathit{\text{regret}}$, develop an online alternating time-averaged gradient method that is capable of leveraging smoothness, and give regret bounds in terms of the path-length of the inner and outer minimizer sequences.