Complexity Classes for Online Problems with and without Predictions

2024-06-26

arXiv.org

AI Summary

This work addresses online optimization problems—both with and without predictions—by establishing the first systematic computational complexity theory framework. Method: Focusing on minimization problems under binary predictions, it introduces an error-measure-driven hierarchy of complexity classes grounded in the canonical hard problem of string guessing; it defines novel error-sensitive reductions, completeness, and hardness notions, and employs information-theoretic analysis, generalized oracle models, and formal online algorithm modeling. Contribution/Results: The paper constructs the first scalable, prediction-augmented complexity hierarchy; proves completeness of classical problems—including paging—within each level of the hierarchy; and uniformly extends known lower bounds across the entire class family. Crucially, it reveals a universal relationship between prediction error magnitude and lower bounds for online problems, thereby unifying and generalizing prior hardness results in the context of predictive assistance.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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 predictions apply directly to all hard problems for each class in the hierarchy based on the canonical pair of error measures. Our work also implies corresponding complexity classes for classic online problems without predictions, with the corresponding complete problems.

Problem

• Develop complexity theory for online problems with predictions
• Define hierarchy of complexity classes for minimization problems
• Analyze classic online problems without predictions complexity

Innovation

• Complexity theory for online predictions
• Binary predictions in minimization problems
• Hierarchies of complexity classes development