Self-optimizing and Pareto-optimal policies in general environments based on Bayes-mixtures

Abstract

The problem of making sequential decisions in unknown probabilistic environments is studied. In cycle t action yt results in perception xt and reward rt, where all quantities in general may depend on the complete history. The perception xt and reward rt 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 ν∈, where is any class of distributions including the true environment μ. We show that the Bayes-optimal policy pξ based on the mixture ξ is self-optimizing in the sense that the average value converges asymptotically for all ν∈ to the optimal value achieved by the (infeasible) Bayes-optimal policy pμ which knows μ in advance. We show that the necessary condition that admits self-optimizing policies at all, is also sufficient. No other structural assumptions are made on . As an example application, we discuss ergodic Markov decision processes, which allow for self-optimizing policies. Furthermore, we show that pξ is Pareto-optimal in the sense that there is no other policy yielding higher or equal value in all environments ν∈ and a strictly higher value in at least one.