A key problem in reinforcement learning is finding a good balance between the need to explore the environment and the need to gain rewards by exploiting existing knowledge. Much research has been devoted to this topic, and many of the proposed methods are aimed simply at ensuring that enough samples are gathered to estimate well the value function. In contrast, [Bellman and Kal-aba, 19591 proposed constructing a representation in which the states of the original system are paired with knowledge about the current model. Hence, knowledge about the possible Markov models of the environment is represented and maintained explicitly. Unfortunately, this approach is intractable except for bandit problems (where it gives rise to Gittins indices, an optimal exploration method). In this paper, we explore ideas for making this method computationally tractable. We maintain a model of the environment as a Markov Decision Process. We sample finite-length trajectories from the infinite tree using ideas based on sparse sampling. Finding the values of the nodes of this sparse subtree can then be expressed as an optimization problem, which we solve using Linear Programming. We illustrate this approach on a few domains and compare it with other exploration algorithms.
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