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We consider a stochastic inventory control problem under censored demands,lost sales, and positive lead times. This is a fundamental problem in inventorymanagement, with significant literature establishing near-optimality of asimple class of policies called ``base-stock policies'' for the underlyingMarkov Decision Process (MDP), as well as convexity of long run average-costunder those policies. We consider the relatively less studied problem ofdesigning a learning algorithm for this problem when the underlying demanddistribution is unknown. The goal is to bound regret of the algorithm whencompared to the best base-stock policy. We utilize the convexity properties anda newly derived bound on bias of base-stock policies to establish a connectionto stochastic convex bandit optimization.  Our main contribution is a learning algorithm with a regret bound of$\tilde{O}(L\sqrt{T}+D)$ for the inventory control problem. Here $L$ is thefixed and known lead time, and $D$ is an unknown parameter of the demanddistribution described roughly as the number of time steps needed to generateenough demand for depleting one unit of inventory. Notably, even though thestate space of the underlying MDP is continuous and $L$-dimensional, our regretbounds depend linearly on $L$. Our results significantly improve the previouslybest known regret bounds for this problem where the dependence on $L$ wasexponential and many further assumptions on demand distribution were required.The techniques presented here may be of independent interest for other settingsthat involve large structured MDPs but with convex cost functions.

Learning in structured MDPs with convex cost functions: Improved regret bounds for inventory management