Hutter M. (2002) Self-Optimizing and Pareto-Optimal Policies in General Environments Based on Bayes-Mixtures. In: Kivinen J., Sloan R.H. (eds) Computational Learning Theory. COLT 2002. Lecture Notes in Computer Science, vol 2375. Springer, Berlin, Heidelberg

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 μ. 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 μ is known. Reinforcement learning is usually used if μ is unknown. In the Bayesian approach one defines a mixture distribution ξ as a weighted sum of distributions \(
\mathcal{V} \in \mathcal{M}
\)
, where \(
\mathcal{M}
\)
is any class of distributions including the true environment μ. We show that the Bayes-optimal policy p^{ξ}based on the mixture ξ is self-optimizing in the sense that the average value converges asymptotically for all \(
\mu \in \mathcal{M}
\)
to the optimal value achieved by the (infeasible) Bayes-optimal policy p^{μ} which knows μ in advance. We show that the necessary condition that \(
\mathcal{M}
\)
admits self-optimizing policies at all, is also sufficient. No other structural assumptions are made on \(
\mathcal{M}
\)
. As an example application, we discuss ergodic Markov decision processes, which allow for self-optimizing policies. Furthermore, we show that p^{λ} is Pareto-optimal in the sense that there is no other policy yielding higher or equal value in all environments \(
\mathcal{V} \in \mathcal{M}
\)
and a strictly higher value in at least one.