Direct image alignment is a widely used technique for relative 6DoF pose estimation between two images, but its accuracy strongly depends on pose initialization.Therefore, recent end-to-end frameworks focused on training objectives, such as the Gauss-Newton loss, which increase the convergence basin of the learned feature descriptors.However, the training data may be biased toward a specific type of motion and pose initialization,thus limiting the generalization of these methods.In this work, we derive a closed-form solution to the expected optimum of the Gauss-Newton loss. The solution is agnostic to the underlying feature representation and allows us to dynamically adjust the basin of convergence according to our assumptions about the uncertainty in the current estimates. This offers effective control over the convergence properties of the algorithm.Despite using self-supervised feature embeddings, our solution achieves compelling accuracy w.r.t. the state-of-the-art direct image alignment methods trained end-to-end with pose supervision, and exhibits improved robustness to pose initialization.Our analytical solution provides insight into the inherent limitations of end-to-end learning with the Gauss-Newton loss and establishes an intriguing connection between direct image alignment and feature-matching approaches.