Learning Unstable Dynamical Systems with Time-Weighted Logarithmic Loss
This work addresses a specific challenge in system identification for unstable dynamics, offering a targeted solution rather than a broad breakthrough.
The paper tackled the problem of learning unstable dynamical systems by identifying that gradient descent with squared-error loss fails due to imbalanced influence of observations over time, and introduced a time-weighted logarithmic loss that effectively enables learning such systems.
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.