Learning Unstable Dynamical Systems with Time-Weighted Logarithmic Loss

TL;DR

This paper introduces a time-weighted logarithmic loss function to improve the training of unstable linear dynamical systems by addressing the imbalance caused by observations at different times during gradient descent.

Contribution

It proposes a novel loss function that stabilizes learning of unstable systems, overcoming limitations of traditional squared-error loss.

Findings

The new loss function enhances convergence in unstable systems.

Observations at different times impact gradient dynamics variably.

The method effectively learns unstable dynamical systems.

Abstract

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.