This paper considers the design of optimal resource allocation policies in wireless communication systems, which are generically modeled as a functional optimization problem with stochastic constraints. These optimization problems have the structure of a learning problem in which the statistical loss appears as a constraint, motivating the development of learning methodologies to attempt their solution. To handle stochastic constraints, training is undertaken in the dual domain. It is shown that this can be done with small loss of optimality when using near-universal learning parameterizations. In particular, since deep neural networks (DNNs) are near universal, their use is advocated and explored. DNNs are trained here with a model-free primal-dual method that simultaneously learns a DNN parameterization of the resource allocation policy and optimizes the primal and dual variables. Numerical simulations demonstrate the strong performance of the proposed approach on a number of common wireless resource allocation problems.