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 hierarchy of complexity classes 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. Our work also implies the same results for classic online problems without predictions.

翻译：随着机器学习的发展，利用预测的算法引起了广泛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机制。尽管关于问题在时间和空间方面的计算复杂性理论已相当成熟，但对于通常不关注时间和空间资源的在线问题，目前所知甚少。研究人员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一个完全不同的问题。我们开创性地为带有预测的在线问题建立了一套复杂性理论框架，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，我们定义了一个复杂度类别层次结构，并发展了归约、类别成员性、困难性以及完备性等概念。我们的框架包含了处理复杂性理论时预期所需的所有工具，并通过分析具有不同特征的问题来展示这些工具的运用。我们的工作也隐含地得出了经典无预测在线问题的相同结论。