Implicit Bias of Policy Gradient in Linear Quadratic Control:
  Extrapolation to Unseen Initial States
Noam Razin Yotam Alexander Edo Cohen-Karlik Raja Giryes Amir Globerson Nadav Cohen
In modern machine learning, models can often fit training data in numerous
ways, some of which perform well on unseen (test) data, while others do not.
Remarkably, in such cases gradient descent frequently exhibits an implicit bias
that leads to excellent performance on unseen data. This implicit bias was
extensively studied in supervised learning, but is far less understood in
optimal control (reinforcement learning). There, learning a controller applied
to a system via gradient descent is known as policy gradient, and a question of
prime importance is the extent to which a learned controller extrapolates to
unseen initial states. This paper theoretically studies the implicit bias of
policy gradient in terms of extrapolation to unseen initial states. Focusing on
the fundamental Linear Quadratic Regulator (LQR) problem, we establish that the
extent of extrapolation depends on the degree of exploration induced by the
system when commencing from initial states included in training. Experiments
corroborate our theory, and demonstrate its conclusions on problems beyond LQR,
where systems are non-linear and controllers are neural networks. We
hypothesize that real-world optimal control may be greatly improved by
developing methods for informed selection of initial states to train on.