Bibliographic Information: Liu, Z., Attias, I., & Roy, D. M. (2024). Sequential Probability Assignment with Contexts: Minimax Regret, Contextual Shtarkov Sums, and Contextual Normalized Maximum Likelihood. arXiv preprint arXiv:2410.03849v1.
Research Objective: This paper aims to characterize the minimax regret of sequential probability assignment with arbitrary hypothesis classes, including those that are nonparametric and sequential, and to identify a general minimax optimal algorithm for this problem.
Methodology: The authors introduce a new complexity measure called the "contextual Shtarkov sum," which generalizes the classical Shtarkov sum to handle contextual information. They then leverage this measure to analyze the minimax regret and derive the optimal prediction strategy.
Key Findings:
Main Conclusions: This work provides a fundamental understanding of sequential probability assignment with contexts by establishing a tight connection between the minimax regret, the contextual Shtarkov sum, and the cNML algorithm.
Significance: This research significantly advances the theoretical understanding of online learning with logarithmic loss, particularly in the challenging setting of arbitrary hypothesis classes. The introduction of the contextual Shtarkov sum and the cNML algorithm provides valuable tools for analyzing and designing online learning algorithms.
Limitations and Future Research: The paper primarily focuses on theoretical analysis. Exploring the computational efficiency of cNML and developing practically viable approximations would be valuable future directions. Additionally, extending the analysis to handle infinite label spaces could further broaden the applicability of these results.
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by Ziyi Liu, Id... at arxiv.org 10-08-2024
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