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In recent years, various metrics have been developed for measuring the similarity of states in probabilistic transition systems (Desharnais et al., 1999; van Breugel & Worrell, 2001a). In the context of Markov decision processes, we have devised metrics providing a robust quantitative analogue of bisimulation. Most importantly, the metric distances can be used to bound the differences in the optimal value function that is integral to reinforcement learning (Ferns et al. 2004; 2005). More recently, we have discovered an efficient algorithm to calculate distances in the case of finite systems (Ferns et al., 2006). In this thesis, we seek to properly extend state-similarity metrics to Markov decision processes with continuous state spaces both in theory and in practice. In particular, we provide the first distance-estimation scheme for metrics based on bisimulation for continuous probabilistic transition systems. Our work, based on statistical sampling and infinite dimensional linear programming, is a crucial first step in real-world planning; many practical problems are continuous in nature, e.g. robot navigation, and often a parametric model or crude finite approximation does not suffice. State-similarity metrics allow us to reason about the quality of replacing one model with another. In practice, they can be used directly to aggregate states.

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We propose a novel approach to constructing predictive models on long complex symbolic sequences. The models are constructed by first transforming the training sequence n-block structure into a spatial structure of points in a unit hypercube. The transformation between the symbolic and Euclidean spaces embodies a natural smoothness assumption (n-blocks with long common suffices are likely to produce similar continuations) in that the longer is the common suffix shared by any two n-blocks, the closer lie their point representations. Finding a set of prediction contexts is then formulated as a resource allocation problem solved by vector quantizing the spatial representation of the training sequence n-block structure. Our predictive models are similar in spirit to variable memory length Markov models (VLMMs). We compare the proposed models with both the classical and variable memory length Markov models on two chaotic symbolic sequences with different levels of subsequence distribution structure. Our models have equal or better modeling performance, yet, their construction is more intuitive (unlike in VLMMs, we have a clear idea about the size of the model under construction) and easier to automize (construction of our models can be done in a completely self-organized manner, which is shown to be problematic in the case of VLMMs). (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

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