VC Theory for Inventory Policies

arXiv:2404.11509v3 Announce Type: replace
Abstract: There has been growing interest in applying reinforcement learning (RL) to inventory management, either by optimizing over temporal transitions or by learning directly from full historical demand trajectories. This contrasts sharply with classical data-driven approaches, which first estimate demand distributions from past data and then compute well-structured optimal policies via dynamic programming. This paper considers a hybrid approach that combines trajectory-based RL with policy regularization imposing base-stock and $(s, S) $ structures. We provide generalization guarantees for this combined approach for several well-known classes in a $T$-period dynamic inventory model, using tools from the celebrated Vapnik-Chervonenkis (VC) theory, such as the Pseudo-dimension and Fat-shattering dimension. Our results have implications for regret against the best-in-class policies, and allow for an arbitrary distribution over demand sequences, which makes no assumptions such as independence across time.
Surprisingly, we prove that the class of policies defined by $T$ non-stationary base-stock levels exhibits a generalization error that does not grow with $T$, whereas the two-parameter $(s, S)$ policy class has a generalization error growing logarithmically with $T$. Overall, our analysis leverages specific inventory structures within the learning theory framework, and improves sample complexity guarantees even compared to existing results assuming independent demands.