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## Self-Optimizing and Pareto-Optimal Policies in General Environments based on Bayes-Mixtures

Author:Marcus Hutter (2002) Comments:15 LaTeX pages Subj-class:Artificial Intelligence; Learning ACM-class:

I.2; I.2.6; I.2.8; F.1.3; F.2 Reference:Proceedings of the 15th Annual Conference on Computational Learning Theory (COLT 2002) ??-??, Springer Report-no:IDSIA-04-02 and cs.AI/0204040 Paper:LaTeX - PostScript - PDF - Html/Gif Slides:PostScript - PDF

Keywords:Rational agents, sequential decision theory, reinforcement learning, value function, Bayes mixtures, self-optimizing policies, Pareto-optimality, unbounded effective horizon, (non) Markov decision processes.

Abstract:The problem of making sequential decisions in unknown probabilistic environments is studied. In cycletactionyresults in perception_{t}xand reward_{t}r, where all quantities in general may depend on the complete history. The perception_{t}xand reward_{t}rare sampled from the environmental probability distribution_{t}. Sequential decision theory tells us how to act in order to maximize the total expected reward, called value, ifis known. Reinforcement learning is usually used ifis unknown. In the Bayesian approach one defines a mixture distributionas a weighted sum of distributionsM, whereMis any class of distributions including the true environment. We show that the Bayes-optimal policypbased on the mixture^{ }is self-optimizing in the sense that the average value converges asymptotically for allMto the optimal value achieved by the (infeasible) Bayes-optimal policypwhich knows^{ }in advance. We show that the necessary assumption thatMadmits self-optimizing policies at all, is also sufficient. No other structural assumptions are made onM. Furthermore, we show thatpis Pareto-optimal in the sense that there is no other policy yielding higher or equal value in^{ }allenvironmentsMand a strictly higher value in at least one.

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## Table of Contents

- Introduction
- Rational Agents in Probabilistic Environments
- Pareto Optimality of policy
p^{ }- Self-optimizing Policy
pw.r.t. Average Value^{ }- Discounted Future Value Function
- Markov Decision Processes
- Conclusions

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@InProceedings{Hutter:02selfopt, author = "Marcus Hutter", title = "Self-Optimizing and {P}areto-Optimal Policies in General Environments based on {B}ayes-Mixtures", series = "Lecture Notes in Artificial Intelligence", volume = "", year = "2002", pages = "", booktitle = "Proceedings of the 15th Annual Conference on Computational Learning Theory (COLT 2002)", publisher = "Springer", url = "http://www.hutter1.net/ai/selfopt.htm", url2 = "http://arxiv.org/abs/cs.AI/0204040", ftp = "ftp://ftp.idsia.ch/pub/techrep/IDSIA-04-02.ps.gz", keywords = "Rational agents, sequential decision theory, reinforcement learning, value function, Bayes mixtures, self-optimizing policies, Pareto-optimality, unbounded effective horizon, (non) Markov decision processes.", abstract = "The problem of making sequential decisions in unknown probabilistic environments is studied. In cycle $t$ action $y_t$ results in perception $x_t$ and reward $r_t$, where all quantities in general may depend on the complete history. The perception $x_t'$ and reward $r_t$ are sampled from the (reactive) environmental probability distribution $\mu$. This very general setting includes, but is not limited to, (partial observable, k-th order) Markov decision processes. Sequential decision theory tells us how to act in order to maximize the total expected reward, called value, if $\mu$ is known. Reinforcement learning is usually used if $\mu$ is unknown. In the Bayesian approach one defines a mixture distribution $\xi$ as a weighted sum of distributions $\nu\in\M$, where $\M$ is any class of distributions including the true environment $\mu$. We show that the Bayes-optimal policy $p^\xi$ based on the mixture $\xi$ is self-optimizing in the sense that the average value converges asymptotically for all $\mu\in\M$ to the optimal value achieved by the (infeasible) Bayes-optimal policy $p^\mu$ which knows $\mu$ in advance. We show that the necessary condition that $\M$ admits self-optimizing policies at all, is also sufficient. No other structural assumptions are made on $\M$. As an example application, we discuss ergodic Markov decision processes, which allow for self-optimizing policies. Furthermore, we show that $p^\xi$ is Pareto-optimal in the sense that there is no other policy yielding higher or equal value in {\em all} environments $\nu\in\M$ and a strictly higher value in at least one.", }

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