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The present work extends the randomized shortest-paths framework (RSP),interpolating between shortest-path and random-walk routing in a network, inthree directions. First, it shows how to deal with equality constraints on asubset of transition probabilities and develops a generic algorithm for solvingthis constrained RSP problem using Lagrangian duality. Second, it derives asurprisingly simple iterative procedure to compute the optimal, randomized,routing policy generalizing the previously developed "soft" Bellman-Fordalgorithm. The resulting algorithm allows balancing exploitation andexploration in an optimal way by interpolating between a pure random behaviorand the deterministic, optimal, policy (least-cost paths) while satisfying theconstraints. Finally, the two algorithms are applied to Markov decisionproblems by considering the process as a constrained RSP on a bipartitestate-action graph. In this context, the derived "soft" value iterationalgorithm appears to be closely related to dynamic policy programming as wellas Kullback-Leibler and path integral control, and similar to a recentlyintroduced reinforcement learning exploration strategy. This shows that thisstrategy is optimal in the RSP sense - it minimizes expected path cost subjectto relative entropy constraint. Simulation results on illustrative examplesshow that the model behaves as expected.

A Constrained Randomized Shortest-Paths Framework for Optimal Exploration