Learning While Repositioning in On-Demand Vehicle Sharing Networks

We consider a network inventory problem motivated by one-way, on-demand vehicle sharing services. Due to uncertainties in both demand and returns, as well as a fixed number of rental units across an $n$-location network, the service provider must periodically reposition vehicles to match supply with demand spatially while minimizing costs. The optimal repositioning policy under a general $n$-location network is intractable without knowing the optimal value function. We introduce the best base-stock repositioning policy as a generalization of the classical inventory control policy to $n$ dimensions, and establish its asymptotic optimality in two distinct limiting regimes under general network structures. We present reformulations to efficiently compute this best base-stock policy in an offline setting with pre-collected data. In the online setting, we show that a natural Lipschitz-bandit approach achieves a regret guarantee of $\widetilde{O}(T^{\frac{n}{n+1}})$, which suffers from the exponential dependence on $n$. We illustrate the challenges of learning with censored data in networked systems through a regret lower bound analysis and by demonstrating the suboptimality of alternative algorithmic approaches. Motivated by these challenges, we propose an Online Gradient Repositioning algorithm that relies solely on censored demand. Under a mild cost-structure assumption, we prove that it attains an optimal regret of $O(n^{2.5} \sqrt{T})$, which matches the regret lower bound in $T$ and achieves only polynomial dependence on $n$. The key algorithmic innovation involves proposing surrogate costs to disentangle intertemporal dependencies and leveraging dual solutions to find the gradient of policy change. Numerical experiments demonstrate the effectiveness of our proposed methods.