Sequential probability assignment and universal compression go hand in hand. We propose sequential probability assignment for non-binary (and large alphabet) sequences with empirical distributions whose parameters are known to be bounded within a limited interval. Sequential probability assignment algorithms are essential in many applications that require fast and accurate estimation of the maximizing sequence probability. These applications include learning, regression, channel estimation and decoding, prediction, and universal compression. On the other hand, constrained distributions introduce interesting theoretical twists that must be overcome in order to present efficient sequential algorithms. Here, we focus on universal compression for memoryless sources, and present precise analysis for the maximal minimax and the average minimax for constrained distributions. We show that our sequential algorithm based on modified Krichevsky-Trofimov (KT) estimator is asymptotically optimal up to $O(1)$ for both maximal and average redundancies. This paper follows and addresses the challenge presented in \cite{stw08} that suggested "results for the binary case lay the foundation to studying larger alphabets".
Towards a Universal Theory of Artificial Intelligence based on Algorithmic Probability and Sequential Decision Theory
决策理论正式解决了不确定世界中理性主体的问题，如果真实的环境概率分布是已知的。Solomonoff的泛归纳理论形式化地解决了未知分布的序列预测问题。我们统一了这两种理论，并给出了强有力的论据，由此产生的通用AIXI模型在任何可计算的环境中表现最佳。AIXI模型的主要缺点是它是不可计算的。为了克服这个问题，我们构造了一个改进的算法AIXI^tl，它仍然优于任何其他时间t和空间l有界代理。AIXI^tl的计算时间为t x 2^l的数量级。
Exploration-Exploitation Tradeoff in Universal Lossy Compression
Viterbi Faculty of Electrical and Computer Engineering, Technion – Israel Institute of Technology（维特比电气工程与计算机科学学院，技术学院–以色列理工学院）；Department of Electrical Engineering-Systems, Tel Aviv University（电气工程-系统系，特拉维夫大学）
