Vulcan: A Monte Carlo Algorithm for Large Chance Constrained MDPs with
  Risk Bounding Functions
  Chance Constrained Markov Decision Processes maximize reward subject to a
bounded probability of failure, and have been frequently applied for planning
with potentially dangerous outcomes or unknown environments. Solution
algorithms have required strong heuristics or have been limited to relatively
small problems with up to millions of states, because the optimal action to
take from a given state depends on the probability of failure in the rest of
the policy, leading to a coupled problem that is difficult to solve. In this
paper we examine a generalization of a CCMDP that trades off probability of
failure against reward through a functional relationship. We derive a
constraint that can be applied to each state history in a policy individually,
and which guarantees that the chance constraint will be satisfied. The approach
decouples states in the CCMDP, so that large problems can be solved
efficiently. We then introduce Vulcan, which uses our constraint in order to
apply Monte Carlo Tree Search to CCMDPs. Vulcan can be applied to problems
where it is unfeasible to generate the entire state space, and policies must be
returned in an anytime manner. We show that Vulcan and its variants run tens to
hundreds of times faster than linear programming methods, and over ten times
faster than heuristic based methods, all without the need for a heuristic, and
returning solutions with a mean suboptimality on the order of a few percent.
Finally, we use Vulcan to solve for a chance constrained policy in a CCMDP with
over $10^{13}$ states in 3 minutes.
