Generalized Dual Decomposition for Bounding Maximum Expected Utility of Influence Diagrams with Perfect Recall

We introduce a generalized dual decomposition bound for computing the maximum expected utility of influence diagrams based on the dual decomposition method generalized to $L^p$ space.  The main goal is to devise an approximation scheme free from translations required by existing variational approaches while exploiting the local structure of sum of utility functions as well as the conditional independence of probability functions.  In this work, the generalized dual decomposition method is applied to the algebraic framework called valuation algebra for influence diagrams which handles probability and expected utility as a pair. The proposed approach allows a sequential decision problem to be decomposed as a collection of sub-decision problems of bounded complexity and the upper bound of maximum expected utility to be computed by combining the local expected utilities. Thus, it has a flexible control of space and time complexity for computing the bound.  In addition, the upper bounds can be further minimized by reparameterizing the utility functions. Since the global objective function for the minimization is nonconvex, we present a gradient-based local search algorithm in which the outer loop controls the randomization of the initial configurations and the inner loop tightens the upper-bound based on block coordinate descent with gradients perturbed by a random noise. The experimental evaluation demonstrates highlights of the proposed approach on finite horizon MDP/POMDP instances.