VC Theory for Inventory Policies
Yaqi Xie Will Ma Linwei Xin
Advances in computational power and AI have increased interest in
reinforcement learning approaches to inventory management. This paper provides
a theoretical foundation for these approaches and investigates the benefits of
restricting to policy structures that are well-established by decades of
inventory theory. In particular, we prove generalization guarantees for
learning several well-known classes of inventory policies, including base-stock
and (s, S) policies, by leveraging the celebrated Vapnik-Chervonenkis (VC)
theory. We apply the concepts of the Pseudo-dimension and Fat-shattering
dimension from VC theory to determine the generalizability of inventory
policies, that is, the difference between an inventory policy's performance on
training data and its expected performance on unseen data. We focus on a
classical setting without contexts, but allow for an arbitrary distribution
over demand sequences and do not make any assumptions such as independence over
time. We corroborate our supervised learning results using numerical
simulations.
  Managerially, our theory and simulations translate to the following insights.
First, there is a principle of "learning less is more" in inventory management:
depending on the amount of data available, it may be beneficial to restrict
oneself to a simpler, albeit suboptimal, class of inventory policies to
minimize overfitting errors. Second, the number of parameters in a policy class
may not be the correct measure of overfitting error: in fact, the class of
policies defined by T time-varying base-stock levels exhibits a generalization
error comparable to that of the two-parameter (s, S) policy class. Finally, our
research suggests situations in which it could be beneficial to incorporate the
concepts of base-stock and inventory position into black-box learning machines,
instead of having these machines directly learn the order quantity actions.