With the developments in machine learning, there has been a surge in interest and results focused on algorithms utilizing predictions, not least in online algorithms where most new results incorporate the prediction aspect for concrete online problems. While the structural computational hardness of problems with regards to time and space is quite well developed, not much is known about online problems where time and space resources are typically not in focus. Some information-theoretical insights were gained when researchers considered online algorithms with oracle advice, but predictions of uncertain quality is a very different matter. We initiate the development of a complexity theory for online problems with predictions, considering minimization problems and one prediction bit per request. Based on the most generic hard online problem type, string guessing, we define a family of hierarchies of complexity classes (indexed by pairs of error measures) and develop notions of reductions, class membership, hardness, and completeness. Our framework contains all the tools one expects to find when working with complexity, and we illustrate our tools by analyzing problems with different characteristics. In addition, we show that known lower bounds for paging with discard predictions apply directly to all hard problems for each class in the hierarchy based on the canonical pair of error measures. This paging problem is not complete for these classes. Our work also implies corresponding complexity classes for classic online problems without predictions, with the corresponding complete problems.

翻译：随着机器学习的发展，利用预测信息的算法引起了广泛关注和研究，特别是在在线算法领域，大多数新成果都将预测维度纳入具体在线问题的研究中。尽管针对时间和空间资源的计算复杂度理论已相当成熟，但关于在线问题中此类资源并非核心关注点的结构性计算难度的研究仍十分有限。虽然研究者通过带神谕建议的在线算法获得了一些信息论层面的见解，但不确定质量的预测却是一个迥然不同的课题。我们开创性地为带预测的在线问题建立了复杂度理论体系，聚焦于最小化问题及每个请求使用单比特预测的场景。基于最通用的难解在线问题类型——字符串猜测，我们定义了一系列复杂度类层级结构（以误差测量对为索引），并发展了归约、类成员、难度和完备性等概念。该框架具备复杂度研究所需的所有工具，并通过分析不同特性的问题来阐释这些工具的应用。此外，我们证明基于预测舍弃的分页问题已知下界可直接适用于规范误差测量对层级结构中每个类的所有难解问题，而该分页问题对于这些类并非完备的。本工作还推导出经典无预测在线问题对应的复杂度类及其相应的完备问题。