Therefore, the weights w are not guaranteed to converge in approximate Q-learning. However, it is possible to modify the weight update in Eq. ( ) in order to yield a suitably convergent learning algorithm. To this end, we modify the Greedy-GQ method of Ref 46, which relies on minimizing not the Bellman error directly, but rather the Bellman error projected onto the basis of feature vectors. This is to compensate the gradient due to the lack of an incomplete basis. These terms can be understood in analogy to the Pulay forces 47 in conventional electronic structure nuclear gradients, where additional terms arise due to the fact that the electronic structure calculations are performed in an incomplete basis.