

![0_image_0.png](0_image_0.png)

![0_image_1.png](0_image_1.png)

the fastest. Since API (LSPI) is not guaranteed to converge, we ran it for at most 20 iterations, which was an upper bound on the number of iterations of OAPI. The results demonstrate that ABP minimizes the L∞ Bellman residual much more consistently than the other methods. Note, however, that all the considered algorithms would perform significantly better given a finer approximation.

## 9. Conclusion And Future Work

We proposed and analyzed approximate bilinear programming, a new value-function approximation method, which provably minimizes bounds on policy loss. ABP returns the optimal approximate value function with respect to the Bellman residual bounds, despite being formulated with regard to transitive-feasible value functions. We also showed that there is no asymptotically simpler formulation, since finding the closest value function and