Analytic Decision analysis via symbolic dynamic programming for parameterized hybrid MDPs

Decision analysis w.r.t. unknown parameters is a critical task in decision-making under uncertainty. For example, we may need to (i) perform inverse learning of the cost parameters of a multi-objective reward based on observed agent behavior; (ii) perform sensitivity analyses of policies to various parameter settings; or (iii) analyze and optimize policy performance as a function of policy parameters. When such problems have mixed discrete and continuous state and/or action spaces, this leads to parameterized hybrid MDPs (PHMDPs) that are often approximately solved via discretization, sampling, and/or local gradient methods (when optimization is involved). In this paper we combine two recent advances that allow for the first exact solution and optimization of PHMDPs. We first show how each of the aforementioned use cases can be formalized as PHMDPs, which can then be solved via an extension of symbolic dynamic programming (SDP) even when the solution is piecewise nonlinear. Secondly, we can leverage recent advances in non-convex solvers that require symbolic forms of the objective function for non-convex global optimization in (i), (ii), and (iii) using SDP to derive symbolic solutions for each PHMDP formalization. We demonstrate the efficacy and scalability of our optimal analytical framework on nonlinear examples of each of the aforementioned use cases.