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, focusing on binary predictions for minimization problems. 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，利用预测的算法受到越来越多的关注并取得了大量成果，尤其是在在线算法领域，大多数新成果都针对具体在线问题融入了预测要素。尽管关于问题在时间和空间方面的结构计算复杂性已有较为成熟的理论，但对于通常不聚焦于时间和空间资源的在线问题，人们所知甚少。研究者们在考虑带有预言机建议的在线算法时获得了一些信息论层面的见解，但质量不确定的预测则是全然不同的问题。我们开创性地为带预测的在线问题建立了一套复杂度理论，重点关注最小化问题的二元预测。基于最通用的困难在线问题类型——字符串猜测，我们定义了一系列复杂度类的层次结构（由误差度量对索引），并发展了归约、类成员、困难性以及完备性的概念。我们的框架包含了处理复杂度时预期应具备的所有工具，并通过分析具有不同特征的问题来展示这些工具的用法。此外，我们证明了基于规范误差度量对的每一层次类别中，所有困难问题的已知下界可直接应用于带丢弃预测的分页问题。该分页问题并非这些类的完备问题。我们的工作也隐含地给出了无预测经典在线问题的对应复杂度类及其相应的完备问题。