  Existing value function approximation methods have been successfully used in
many applications, but they often lack useful a priori error bounds. We propose
a new approximate bilinear programming formulation of value function
approximation, which employs global optimization. The formulation provides
strong a priori guarantees on both robust and expected policy loss by
minimizing specific norms of the Bellman residual. Solving a bilinear program
optimally is NP-hard, but this is unavoidable because the Bellman-residual
minimization itself is NP-hard. We describe and analyze both optimal and
approximate algorithms for solving bilinear programs. The analysis shows that
this algorithm offers a convergent generalization of approximate policy
iteration. We also briefly analyze the behavior of bilinear programming
algorithms under incomplete samples. Finally, we demonstrate that the proposed
approach can consistently minimize the Bellman residual on simple benchmark
problems.
