  When training the parameters of a linear dynamical model, the gradient
descent algorithm is likely to fail to converge if the squared-error loss is
used as the training loss function. Restricting the parameter space to a
smaller subset and running the gradient descent algorithm within this subset
can allow learning stable dynamical systems, but this strategy does not work
for unstable systems. In this work, we look into the dynamics of the gradient
descent algorithm and pinpoint what causes the difficulty of learning unstable
systems. We show that observations taken at different times from the system to
be learned influence the dynamics of the gradient descent algorithm in
substantially different degrees. We introduce a time-weighted logarithmic loss
function to fix this imbalance and demonstrate its effectiveness in learning
unstable systems.
