This paper investigates the general problem of resource allocation for mitigating channel fading effects in Free Space Optical (FSO) communications. The resource allocation problem is modeled as the constrained stochastic optimization framework, which covers a variety of FSO scenarios involving power adaptation, relay selection and their joint allocation. Under this framework, we propose two algorithms that solve FSO resource allocation problems. We first present the Stochastic Dual Gradient (SDG) algorithm that is shown to solve the problem exactly by exploiting the strong duality but whose implementation necessarily requires explicit and accurate system models. As an alternative we present the Primal-Dual Deep Learning (PDDL) algorithm based on the SDG algorithm, which parametrizes the resource allocation policy with Deep Neural Networks (DNNs) and optimizes the latter via a primal-dual method. The parametrized resource allocation problem incurs only a small loss of optimality due to the strong representational power of DNNs, and can be moreover implemented without knowledge of system models. A wide set of numerical experiments are performed to corroborate the proposed algorithms in FSO resource allocation problems. We demonstrate their superior performance and computational efficiency compared to the baseline methods in both continuous power allocation and binary relay selection settings.