Marrying Stochastic Gradient Descent with Bandits : Learning Algorithms for Inventory Systems with Fixed Costs

We consider a periodic-review single-product inventory system with fixed cost under censored demand. Under full demand distributional information, it is well-known that the celebrated $(s,S)$ policy is optimal. In this paper, we assume the firm does not know the demand distribution a priori, and makes adaptive inventory ordering decision in each period based only on the past sales (a.k.a. censored demand) data. The standard performance measure is regret, which is the cost difference between a feasible learning algorithm and the clairvoyant (full-information) benchmark. Compared with prior literature, the key difficulty of this problem lies in the loss of joint convexity of the objective function, due to the presence of fixed cost. We develop a nonparametric learning algorithm termed the $(\delta, S)$ policy that combines the powers of stochastic gradient descent, bandit controls, and simulation-based methods in a seamless and non-trivial fashion. We prove that the cumulative regret is $O(\log T\sqrt{T})$, which is provably tight up to a logarithmic factor. We also develop several technical results that are of independent interest. We believe that the framework developed could be widely applied to learning other important stochastic systems with partial convexity in the objectives