Application of the Mirror Descent Method to minimize average losses coming by a poisson flow

We treat a convex problem to minimize average loss function for a stochastic system operating in continuous time. The losses on time horizon T arise at the jump times of a Poisson process with intensity being an unknown random process. The oracle gives randomly noised gradients of the loss function; the noises are additive, unbiased, with the bounded dual norm in average square sense. The goal consists in minimizing the average integral loss over a given convex compact set in the N-dimension space. We propose a mirror descent algorithm and prove an explicit upper bound for the average integral loss regret. The bound is of type “square root of T” with an explicit coefficient. Finally, we describe an example of optimization for a server processing a stream of incoming requests, and we discuss simulation results.