  We investigate the convergence and stability properties of the decoupled
extended Kalman filter learning algorithm (DEKF) within the long-short term
memory network (LSTM) based online learning framework. For this purpose, we
model DEKF as a perturbed extended Kalman filter and derive sufficient
conditions for its stability during LSTM training. We show that if the
perturbations -- introduced due to decoupling -- stay bounded, DEKF learns LSTM
parameters with similar convergence and stability properties of the global
extended Kalman filter learning algorithm. We verify our results with several
numerical simulations and compare DEKF with other LSTM training methods. In our
simulations, we also observe that the well-known hyper-parameter selection
approaches used for DEKF in the literature satisfy our conditions.
