  A key limitation in using various modern methods of machine learning in
developing feedback control policies is the lack of appropriate methodologies
to analyze their long-term dynamics, in terms of making any sort of guarantees
(even statistically) about robustness. The central reasons for this are largely
due to the so-called curse of dimensionality, combined with the black-box
nature of the resulting control policies themselves. This paper aims at the
first of these issues. Although the full state space of a system may be quite
large in dimensionality, it is a common feature of most model-based control
methods that the resulting closed-loop systems demonstrate dominant dynamics
that are rapidly driven to some lower-dimensional sub-space within. In this
work we argue that the dimensionality of this subspace is captured by tools
from fractal geometry, namely various notions of a fractional dimension. We
then show that the dimensionality of trajectories induced by model free
reinforcement learning agents can be influenced adding a post processing
function to the agents reward signal. We verify that the dimensionality
reduction is robust to noise being added to the system and show that that the
modified agents are more actually more robust to noise and push disturbances in
general for the systems we examined.
