Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs

We generalize the belief-propagation algorithm to sparse random networks with arbitrary distributions of motifs (triangles, loops, etc.). Each vertex in these networks belongs to a given set of motifs (generalization of the configuration model). These networks can be treated as sparse uncorrelated hypergraphs in which hyperedges represent motifs. Here a hypergraph is a generalization of a graph, where a hyperedge can connect any number of vertices. These uncorrelated hypergraphs are tree-like (hypertrees), which crucially simplify the problem and allow us to apply the belief-propagation algorithm to these loopy networks with arbitrary motifs. As natural examples, we consider motifs in the form of finite loops and cliques. We apply the belief-propagation algorithm to the ferromagnetic Ising model on the resulting random networks. We obtain an exact solution of this model on networks with finite loops or cliques as motifs. We find an exact critical temperature of the ferromagnetic phase transition and demonstrate that with increasing the clustering coefficient and the loop size, the critical temperature increases compared to ordinary tree-like complex networks. Our solution also gives the birth point of the giant connected component in these loopy networks.

Approximate Policy Iteration with a Policy Language Bias: Solving Relational Markov Decision Processes

We study an approach to policy selection for large relational Markov Decision Processes (MDPs). We consider a variant of approximate policy iteration (API) that replaces the usual value-function learning step with a learning step in policy space. This is advantageous in domains where good policies are easier to represent and learn than the corresponding value functions, which is often the case for the relational MDPs we are interested in. In order to apply API to such problems, we introduce a relational policy language and corresponding learner. In addition, we introduce a new bootstrapping routine for goal-based planning domains, based on random walks. Such bootstrapping is necessary for many large relational MDPs, where reward is extremely sparse, as API is ineffective in such domains when initialized with an uninformed policy. Our experiments show that the resulting system is able to find good policies for a number of classical planning domains and their stochastic variants by solving them as extremely large relational MDPs. The experiments also point to some limitations of our approach, suggesting future work.