  We seek the best traffic allocation scheme for the edge-cloud computing
network that satisfies constraints and minimizes the cost based on burstable
billing. First, for a fixed network topology, we formulate a family of integer
programming problems with random parameters describing the various traffic
demands. Then, to overcome the difficulty caused by the discrete feature of the
problem, we generalize the Gumbel-softmax reparameterization method to induce
an unconstrained continuous optimization problem as a regularized continuation
of the discrete problem. Finally, we introduce the Gumbel-softmax sampling
network to solve the optimization problems via unsupervised learning. The
network structure reflects the edge-cloud computing topology and is trained to
minimize the expectation of the cost function for unconstrained continuous
optimization problems. The trained network works as an efficient traffic
allocation scheme sampler, remarkably outperforming the random strategy in
feasibility and cost function value. Besides testing the quality of the output
allocation scheme, we examine the generalization property of the network by
increasing the time steps and the number of users. We also feed the solution to
existing integer optimization solvers as initial conditions and verify the
warm-starts can accelerate the short-time iteration process. The framework is
general with solid performance, and the decoupled feature of the random neural
networks is adequate for practical implementations.
