Online optimization is a well-established optimization paradigm that aims to make a sequence of correct decisions given knowledge of the correct answer to previous decision tasks. Bilevel programming involves a hierarchical optimization problem where the feasible region of the so-called outer problem is restricted by the graph of the solution set mapping of the inner problem. This paper brings these two ideas together and studies an online bilevel optimization setting in which a sequence of time-varying bilevel problems are revealed one after the other. We extend the known regret bounds for single-level online algorithms to the bilevel setting. Specifically, we introduce new notions of bilevel regret, develop an online alternating time-averaged gradient method that is capable of leveraging smoothness, and provide regret bounds in terms of the path-length of the inner and outer minimizer sequences.

翻译：在线优化是一种成熟的优化范式，旨在根据先前决策任务正确答案的知识，做出连续正确的决策。双层规划涉及一个层次优化问题，其中所谓外部问题的可行域受限于内部问题解集映射的图。本文将这两种思想结合起来，研究了一个在线双层优化设置，其中一系列随时间变化的双层问题逐一呈现。我们将已知的单层在线算法遗憾界扩展到了双层设置。具体而言，我们引入了双层遗憾的新概念，开发了一种能够利用平滑性的在线交替时间平均梯度方法，并提供了关于内部和外部最小化序列路径长度的遗憾界。