We study the finite-horizon Restless Multi-Armed Bandit (RMAB) problem with $N$ homogeneous arms. Prior work has shown that when an RMAB satisfies a non-degeneracy condition, Linear-Programming-based (LP-based) policies derived from the fluid approximation, which captures the mean dynamics of the system, achieve an exponentially small optimality gap. However, it is common for RMABs to be degenerate, in which case LP-based policies can result in a $\Theta(1/\sqrt{N})$ optimality gap per arm. In this paper, we propose a novel Stochastic-Programming-based (SP-based) policy that, under a uniqueness assumption, achieves an $\tilde{\mathcal{O}}(1/N)$ optimality gap for degenerate RMABs. Our approach is based on the construction of a Gaussian stochastic system that captures not only the mean but also the variance of the RMAB dynamics, resulting in a more accurate approximation than the fluid approximation. We then solve a stochastic program for this system to obtain our policy.

In this paper, we consider a multi-stage dynamic assortment optimization problem with multi-nomial choice modeling (MNL) under resource knapsack constraints. Given the current resource inventory levels, the retailer makes an assortment decision at each period, and the goal of the retailer is to maximize the total profit from purchases. With the exact optimal dynamic assortment solution being computationally intractable, a practical strategy is to adopt the re-solving technique that periodically re-optimizes deterministic linear programs (LP) arising from fluid approximation. However, the fractional structure of MNL makes the fluid approximation in assortment optimization highly non-linear, which brings new technical challenges. To address this challenge, we propose a new epoch-based re-solving algorithm that effectively transforms the denominator of the objective into the constraint. Theoretically, we prove that the regret (i.e., the gap between the resolving policy and the optimal objective of the fluid approximation) scales logarithmically with the length of time horizon and resource capacities.

Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous state-space endowed with a dynamics for the approximating process. We construct here a general method based on spectral analysis of the transition matrix of the CTMC, without the need for a population structure. Specifically, we use the popular manifold learning method of diffusion maps to analyse the transition matrix as the operator of a hidden continuous process. An embedding of states in a continuous space is recovered, and the space is endowed with a drift vector field inferred via Gaussian process regression. In this manner, we construct an ODE whose solution approximates the evolution of the CTMC mean, mapped onto the continuous space (known as the fluid limit).